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6CBIPT0RUM «RAECOKUM ET EOMANORUM 
TEUBNERIANA. 



EUCLIDIS 

lO P E R A M N I A. 

I. L. HEIBERG i;t H. MENGE. 

EUCLIDIS ELEMENTA. 

UDIurr E'l' LATINl': lN-lKHl'ltBrATU8 EST 

IJL. HEIBERG. 

iJOL. i Linitos 1-iv criN'nNKN8. 



LH-SIAE 

tS AEUIUUS a. Q.TEUIINERI. 



r 

I 
I 



BIBLI 

SCEIPTORnll GRAECORIH 



OTHECA 

ET KOJIASORIII TEDBSERU» 



iDiliwldea ed. Blafs. £d. n 

Antlilaiu ed. Ro>t 

Antholagli latlurt ti.Riat. . 



n ed. Jlla/i. Ed. n. 



- pliMici Bd.PrimH, . . 

— Kthlei NlfiDnache* ed.Si 

dii wnlo ete. od, Prunt 



rlBnl cipcditlo « 



:l<l fineei e 

inie-, kplt. 



Bd. Fiiper. . I 



- od. I 



•oll. . 



lie liKlio Okllieii 

dllil. Kd. min. .... 

raanlii FflllK ad. Roa 

"«tullas ed. MilHr 

Jilnlii», TibnllH, FrapeTtlne . 
Cehetli tihali ei" '- '■- 



L Droaii:! 



WeKnlierg. StdIL a,a 






s Phrf gliH ed. JTniJ 



lor/. 






Dlctji CretenHl 
DliiirehNi ed. 1 
Dio ChiIiiii ed. DiitUorf. I 
Dlo Chryeogl. ed. DindBT/. 
Dlod. SieDlin ed. Bindarf. 
DIOBielDe ed. Kiefiiinii. i 
DruoBlliu ed. dtLuhn. . 
Kclogae poet. litli. ed. B 



Grattci iicript ed. i 

m. Frig 

Eniehtii) ed. DiHdorf. 1 toI 
EitroplBi ed. OitlKA. . . . 
Fabgiie leiopltne ed. Mabit 



FrOiitlDiis ed. Zi«Irric* . 
miiH ed. HukUi . . . 
fielllD* ed. BtrU. i *oL 
UelladDr ed. SbUit . . 
nerodiiD ed. BiUir . . 
Heradol.Bt ed. Bieiidi. i 



BnjtMn» Xileili» ed. 
nieronlunB ed. EerOliig 
UletDrlcl (iraecl nlDOre 

lorli Apo 



lilelorfi 



kpollaail ed, fliiu , 
, kpll. 1 Bud 



Biini 



1 (Hlii n . 



s liiuini. 



n. 1 (Odjiei 

- — — n. 3 (Odjruei n) . 

nontliii ed. ifvi/T 

Hjglnl Oromltlci llber ed. G, 
Hjmnl Hanerici ed. fiaumMiir 
Hjperld» ed. JBIaf; Bd. n. , 



lliidli 
ineertl in 

laaeph». FUrlnii^ ed. Bt 



iriBl 






lMCriteiedd,finHf(retfi/a/i. traU. 2,Ti 



EUCLIDIS 



OPERAOMNIA. 



EDIDERUNT 



I. L. HEIBERG ET H. MENGK 




LIPSIAE 

IN AEDIBUS B. G. TEU3NERI. 
MDCCCLXXXm. 



EUCLIDI8 



E L E M E N T A. 



EDIDIT ET IiATINB INTERPRBTATUS B8T 



I. L. HEIBER6, 

DB. PBIL. 



UOL. I. 

LIBROS I— IV CONTINENS. 




LIPSIAE 

IN AEDIBUS B. G. TEUBNERI. 
MDCCCLXXXin. 




LIFSIAB : TTPI8 B. O. TXUBMaBX. 




£lemeiita EucUdia paene per tria saecula pro fun- 
dameuto critica aolam editionem principem habuerant, 
quae prodiit Basileae a. 1533; nam Gregorius iu ele- 
mentis totus fere ab illa editione pendet. quod fun- 
dameotum quale fuerit, inde intellegitur, qaod editio 
Basileenaia pro conauetudine illius temporis ad fidem 
pancissimorum nec optimorum codicum facta est, 
cam tamen elemeutorum tot exsteut codices autiquia- 
simi et praeatantisaimi, quot haud facile cuiuaquam 
scriptoris Oraeci. itaque initio nostri saeculi Peyrar- 
du8 optime de elementia meritus est, quod unum sal- 
tem codicem antiquum et eum omnium praeatantissi- 
mum, quippe qui recenaionem Tbeone antiquiorem 
contineret, in editione Basileensi emendanda adliibuit. 
honc codicem e latebris Uaticanis protraxiaae prae- 
stantiamque eiua agnouisae, gloria est Peyrardi baud 
parui aestimanda. sed neque ubique recto firmoque 
iudicio in uera scriptura eligenda usus est, in primis 
quia bonis codicibua receosionis Tbeonis caruit, neque 
inaentum suum tenuit recteque aestimauii buc adce- 
dit, *quod editio eius et inhabilis et his temporibus 
perrara eat; nee ii, qui post Pejrardum eleuenta edi- 
derunt, subsidia critica auxerunt neque omnino rem 



VI PRAEFATIO. 

ita egerunt, ut textus elementorum Batis certo et ad 
usum prompto fundamento niti uideri possit. de cete- 
ris acriptie Euclidis multo etiam peius actura esse, 
satts constat. 

Qaae cum a multis intellegi uiderem, Arctimedi 
Euclidem adiongere constitui, et ut hunc laborem, 
quem iara diu animo uoluebam, tandem aHquandd 
suseiperem, eo magis impellebar, quod editionem . 
cMmedis ab homioibus doctis beneuolenter adcipi, 
erroribns, quos in priraitiis illis uitarc non potuiseexd 
indulgeri uidebam, et usu edoctum me iam melioH 
praeatare posae sperabam. 

Sed statim apparuit, neque res rationesque ueqoj 
uires raeas toti operi, quod mihi proposueram, 
cere. tot codicea conferendi erant, tot bibliothecae i 
neribua Jonginquis adeundae. itaque Henricum Meng< 
u. d., quem sciebam et ipsum in Euclide occupatui 
esse, interrogaui, uelletne partem operis suscipei 
adnuit, et ita inter nos comparatum est, ut ille Data, 
Phaenomena, acripta musiea, ego Elementa, Optica, 
Catoptrica ederem, et ut codices conjuncta opera con- 
ferremu3. sed sic quoque in elementia e magna eopia 
sabsidioruni pauea eligere coactus eura. nara cum uiz 
ulla sit minima bibliotheca, in qua nou adaeruetur 
codex aliquis eleraentorum, inde ab initio de onmibua 
codieibuB conferendis aut cette inBpiciendis desperan- 
dum erat. uellem equidem licuisset pturibus codicibua 
uti, sed ut aliquo tamen modo paucia, quoB contnli, 
contenti esse poBsimus, facit et aingularis ratio, qua 
nobis tradita sunt elementa EucHdis, et uetuataa et 
boQitas codicum a me uaurpatorum. nam satis notum 



PRAEFATIO. Vn 

esty plerosque omnes codices e recensione Theonis flu- 

xisse, et Uaticanum Peyrardi solum fere antiquiorem 

formam seruasse. quem fructum ex hoc casu singu- 

lari capere liceat, et quam rationem critices factitan- 

dae inde sequi putem, pluribus exposui in libro, qui 

inscribitur Studien iiber Euklid p. 177 sq. hoc quidem 

statim adparuit^ primum omnium codicem Uaticanum, 

e qao Peyrardus ea sola enotauerat, quae ei memo- 

rabilia uidebantur, quamuis ipse aliter praedicet, de- 

noo diligenter esse conferendum et praeterea ex reli- 

qois codioibus tantum numerum, ut ueri similiter de 

scriptora Theonis iudicari posset qua in re codices 

Bodleianum, Laurentianum^ Uindobonensem sufficere 

putaui, praesertim cum animaduerterem, eos a palim- 

psesto codice saeculi YII uel YIII; qui in Museo Bri- 

tannico adseruatur, non admodum discrepare. hos co- 

dices pro fundamento habui; sed ad eos in partibus 

quibusdam operis alii adcesserunt et, ut spero, adce- 

dent, uelut in hoc primo uolumine Parisinus quidam 

et in primo libro Bononiensis. hunc ne totum con- 

ferrem^ prohibuerunt temporis angustiae^ sed spes mihi 

est, me breui partem reliquam conferre posse; nam 

in libris stereometricis hic codex maximi momenti 

est. de ceteris subsidiis nouis, sicut de codicibus 

operum minorum, in praefationibus singulorum uolu- 

minum dicetur. 

Confiteor igitur fieri posse, ut inter codices nou- 
dum coUatos lateat thesaurus aliquis (neque enim 
omnes recentiores sunt nec recentiores semper sper- 
nendi), qui mea subsidia uel aequet uel etiam superet. 
sed cum non maxime sit ueri simile, haec, qualiacun- 



Vm PRAEFATIO. 

que suiit, nuDC edere malui, quam opua in infinitum 

differre. 

De consilio meo satis dictum. de forma ac specie 

editionis sufficit conmemorare, eandem me secutum 
esse quam iii Arcliimede edeudo. nam quamquam ui- 
debam, Latinam interpretationem meam a nonuullia 
improbari, tamen hic quoqne Latinam Francogallicae 
Germanaeue aut nulli praetuli; nam interpretationem 
mathematici flagitant, et Latina a pluribua legi potest. 
praeterea rea ipsae tritiorea interpretandi molestiam 
leuiorem reddunt in Euclide quam in Archimede. 
notas perpaucas addidi, quia perpaucia in Euclide 
discentibus consulenti opus eet, si solam intellegentiam 
uerborum tenorisque demonstrationis speetes. nam 
commentarium, cuius hic quoque ingena est materia, 
scribere nolui. quarto uolumini copiosiora prolego- 
mena praemittentur, quibus historia textua elemento- 
rum illuatrabitur. eodem congeram, quae de subsicliis 
deterioribua coUegij nam perapicuitatis causa ea ab 
adparatu critico removenda erant, in quo iis tantum 
codicibus usus aum, quos supra commemoraui. 
his litteria aignifioaui: 
P — cod. Uatican. Gr. 190 Peyrardi saec. X, mem,-J 
bran. bic illic manus recentissima litteraa tem 
pore euanidas renouauit, quam littera ;r signi-l 
ficaui, ubi parum recte acripturam antiquam red-J 
dere uidebatur. libroa IV — IX ipse contuli Ro-B 
mae 1881, librum 11 et partem tertii Mengiu 
primum et reliquam partem tertii Augustuw 
Mau u. d. beneuolenter conferenda suscepit. 
B — cod. Bodleian, Duruillian. X, 1 inf. 2, 30, scr. a.1 



rtH^k 



PRAEFATIO. IX 

888, membran. libros I — VII ipse contuli Oxo- 
niae 1882. 

F — cod. Florentin. Laurentian. XXVIII, 3 saec. X, 
membran. in hoc quoque codice scriptura an- 
tiqua saepe manu saeculi XVI renouata est, 
quae eadem multa folia foliorumue partes re- 
sarcinauit et ultimam partem codicis totam sup- 
pleuit. eam significaui littera % ubicunque an- 
tiquam scripturam uel uitiauit uel ita obscura- 
uit, ut dignosci non posset. totum codicem ipse 
contuli Florentiae 1881. 

V — cod. Uindobon. Gr. 103 saec. XI— XII, membran. 
partem ultimam in charta bombycina suppleuit 
manus saeculi XUI. totum contuli ipse Hauniae 
1880. 

b — cod. bibliothecae communalis Bononiensis nume- 
ris 18-19 signat., saec. XI, membran. librum I 
contuli et alios nonnullos locos inspexi Floren- 
tiae 1881. 

p — cod. Parisin. Gr. 2466 saec. XII, membran. 
librum I contuli Parisiis 1880, libros II— VII 
Ilauniae 1882. 



Restat, ut grato of&cio fungar iis uiris gratias 
quam maximas agendi, qui labori meo fauerunt. pri- 
mum ut itinera Parisios et in Italiam toties facere 
possem, effectum est eximia liberalitate summi Mi- 
nisterii, quod cultui scholisque nostris praeest, et 
instituti Carlsbergici, litteras scientiamque lar- 
giter adiuuantis. etiam praefectis bibliothecarum Uin- 

Euclides, edd. Heiberg et Menge. a** 



X PRAEFATIO. 

dobonensis^ Parisinae^ Bononiensis plurimum 
debeo, quod codices a se adservatos meum in usum 
alio transmitti siuerunt, item praefectis bibliothecae 
regiae Hauniensis et bibliothecae Laurentianae, 
quibus intercedentibus hunc fauorem adeptus sum. 
Carolo Graux, quocum magnam partem itineris 
Italici a. 1881 communiter feci, et qui me in codicum 
aetatibus definiendis ceterisque rebus palaeographicis, 
in quibus cedebat nemini, egregie adiuuabat, quomi- 
nus hoc loco gratias debitas agerem, prohibuit fatum 
nobis amicis eius superstitibus scientiaeque iniquis- 
simum. 

Scr. Hauniae mense Aprili MDCCCLXXXIII. 



STOIXEIA. 



Suolidef, edd. Heiberg et Menge. 



a . 



a . £rifi€t6v i6tLV, ov (leQog ov^iv. 
fi\ rQafi^ri d^ ^^xog aytXatig. 
y . FQafifi^g d^ niQata 6fifi€ta. 
d\ Ev^sta yQafi^i^ i^tiv^ r^tig i^ t6ov totg itp* 
5 iavt^g 6ri(i£L0Lg xettai. 

b\ 'Emq^dveia di i6tvv, o fi^xog xal nXatog fio- 

VOV i%BL. 

g'. ^EatKpavsCag 8\ niQata yQafifiai. 
{;'. ^EnlnsSog imtpavsia i6tiv^ r^tig i^ t6ov tatg 
10 ifp* iavt^g svd^eCaig xettai, 

r( , ^ExCjteSog Sh ymvCa i6tlv ^ iv ininiSip Svo 
yQafificiv antoiiivcnv dkkiqXaiv xal /t^ in ev^eCag xbl- 
fiivwv nQog dXki^kag tAv yQa^ficSv xXC6ig. ^ 

d"'. "Otav S\ at neQcixov6aL trjv yoDvCav yQa^fial 
15 evd^etac m6iv^ evd^Qafifiog xakettai ij yavCa. 

i . "Otav S\ evd^eta in ev%etav 6ta^et6a tdg iip- 

1. Hero def. 2. Ammouius in categ. p. 43. 66. Psellus p. 34. 
cfr. Philoponus iii phys. fol. 6'. Martianus Capella YI, 708. 
Boetius p.d74, 1. 2. Sextus Emp. p. 466,27. 470,24. 704,28. 
Hero def. 3. Philoponus in phys. fol. 6'. Ammonius in cat. 
p. 66. Martianus Capella VI, 708. Boetius p. 374, 2. 3. Boe- 
tias p. 374,3. 4. Hero def. 5. Sextus Emp. p. 716, 28. 717, 

10. Philoponus in anal. 11 fol. 4^, fol. 15. Psellus p. 34. Boe- 
tius p. 374, 5. 5. Hero def. 9. Boetius p. 374, 6. 6. Boe- 
tius p. 374, 7. 7. Hero def. 11. Psellus p. 35. Boetius p. 

374, 7. 8. Hero def. 16. Psellus p. 35. cfr. Sextas Emp. 

p. 718, 12. Boetius p. 874, 10. Martianws Capella VI, 710. 



I. 

Definitiones. 

L Ponctum est^ cuius pars nuUa est. 

n. Linea autem sine latitudine longitudo. 

UI. Lineae autem extrema puncta. 

IV. Recta linea est^ quaecunque ex aequo punctis 
in ea sitis iacet. 

y. Superfides autem est, quod longitudinem et 
latitudinem solum habet. 

YI. Superficiei autem extrema lineae sunt. 

YII. Plana superficies est^ quaecunque ex aequo 
rectis in ea sitis iacet 

yni. Planus autem angulus est duabus lineis in 
plano se tangentibus nec in eadem recta positis alte- 
rius lineae ad alteram inclinatio. 

IX. Ubi uero lineae angulum continentes rectae 
sunt, rectilineus adpellatur angulus. 

X. Ubi uero recta super rectam lineam erecta 

9. Hero def. 17. Boetius p. 874,12. 10. Hero def. 19. Am- 
monins in categ. p. 58. Simplicins in Aristot. de coelo fol. 131^. 
Philoponu8 in phys. i IHI, in anal. H fol. 28^, p. 65. Psellus 
p. 36. Martianus Capella VI, 710. Boetius p. 374, 14. 



NumeroB definitionum om. PFBb. 1. ov6iv F, Paellus, 

Ammonius p. 66. 6. ^x^i fiovov B. 11 ds] supra comp. 

scriptum b. inmiScp'] ininsdog n. 13. Ante ngog ras. 

unius litterae PF. 14. di] d' B. riiv ytavCav nsgiixovaai, 

Proclus; tijv BlqriyLivriv ytovlav P. 15. 7] ycovia naXsLtaL 
Proclus. 



4 2T0IXEIiiS o'. 

aS^S yaviaq Caag KAiijAaig ^oiij, oQ&ii ixKriga i 
fffrav ytovt.mv isri, xal ^ itpeSTrjKvta svQsta xcc^ett 
xttlfttat, i<p' ^v ifpiffzi]xtv. 

la. 'j4fi^XEia yoivitt iszlv rj fiei^av op^^s. 
5 (j5'. 'O^Eta Ss i} ilduoav op&^s. 

ly'. "Opog ieriv, o xivos i^ti xiQag. 

id'. £xW^ ^^^' '^" ^'"' Ttfog ^ rtvov opra 

[e', Kvxios ifti ^XVl''^ iitiiteSov vno iiias yQttfim 
10 [lijg nBQiExo^Bvov [j; xaleltai ns^itpiQBia'] , xqos i 
«qs' ivog ffijftstov Tt5i' ^vros tou ff^fjjftatog xeifiiva 
3ta0ai at X0o3itijttov0«i Bv&Btat [jTpog tjjv rotJ i 
jtAow jTsptqoEpftftv] /'ffat «iAijAats siaiv. 

is'. KivzQov Se tov xi^xAov to ffijfiEfov xaA^m 

16 (£'. itfia'fi£Tpos 5i rou xux^ov ^ffrlv eu&fEa ■ 

5ta Toti xivtQov TiyiLBvyj xaX nEQOTOvfiivij itp' Bxata^ 

%a fliQf] V7t6 tijs tOV XVXloV TtEQKpBQBiag , ^Ttg ] 

dt;i;« tiftvet tov xvxXov. 

H)'. 'HlUXVXkiOV Si iOtl t6 7tBQtB%6{».BVOV ffx^f 

ao rBo TE T^s Sia^itQQV xal tijs dnoXaii^avoniv7}s i 



11. Hero def. 31. Amnioniua in categ. p. 68. pBellufl p. 
MartiftnQs Capella TI, 710. Boetioa p. 374, 18. 12. Hero ( 
20. AmmOQius 1. c. PBellua 1. c. Martiaiiufl Capella 1. c. Boet 
p. 874, 19. 13. PhiioponuB in Aristot. de anima fol. i 

Martianua Capelta VT, 710. Boetiiia p. 374,22. 14. Hero i 
25. Schol. in Hermog. VU' p. 903. cfe. Philop. ad AriBtot. 
anim. h. 7. Martianus Capella VI, 710, BoetiuB p. 374,21. 
15. Hero def. 29. Tanrus apud Philop. in Proclum VI, 21. S 
tna Emp. p. 719,16. Philopon. in aoal 11 fol 88'' " ' ' 



Peellus p. 38. Martianus Capeila TI, 710. Boet 
p. '376,3. 16. PneliUB p. 3B. Martjanus Capella VI,711. Bo 
tiua p. 375.6. 17. Hero def. 30. PbeIIub p. 38. Martianua 

Capella VI, 711. Boetius p. 376, 7. 18. Hero def. 31, Mart, 
Capella VI,711, Boetius p, S76, 12, 



^^ 



J 



ELEBfENTORUM UBER I. 5 

angaloa deinceps positos inter se aeqnales efficit^ rec- 
tas est uterque angulus aequalis^ et recta linea 
erecta perpendicularis adpellatur ad eam, super quam 
erecta esb 

XI. Obtusus angulus est^ qui maior est recto. 

Xn. Acutus ueroy qui minor est recto. 

Xm. Terminus est^ quod alicuius rei extremum est. 

XIY. Figura est^ quod aliquo uel aliquibus ter- 
minis comprehenditur. 

XY. Circulus est figura plana una linea compre- 
hensa^ ad quam quae ab uno puncto intra figuram 
posito educuntur rectae omnes aequales sunt. 

XYI. Centrum autem circuli punctum illud adpel- 
latur. 

XVII. Diametrus autem circuli recta quaedam est 
linea per centrum ducta et terminata utrimque am- 
bitu circuliy quae quidem linea circulum in duas par- 
tes aequales diuidit. 

XVin. Semicirculus autem ea est figura, quae 

1. ^Q^ri iariv SKaxiQa omisso iati, lin. 2 BFV, Simplicius, 
PhilopoDus in anal. II p. 65, Psellus. scripturam receptam 
praebent Pbp, Proclus, Hero, Ammonius, Philoponus in phys. 
i JIU. cfr. prop. 11,12. 2. i^aatv] om.Ammonius, Philoponua 
in phys. 1. c, rsellus, Martianus Capella, Campanus. sv^sta] 
yQafiful Proclus, BV; om. Ammonius. Den. XI— XII permu- 
tant Hero et Ammonius. 6. ty'] t^' V et sic deinceps. 
Deff. Xin— XIV permutat Boetius. 7. iau] Si Fbp. 10. 
ij TtaXsiTat, nsQKpsQsial om. Proclus, Taurus, Sextus Emp., Phi- 
loponus, Boetius; habent praeter codd. Hero, Psellus, Uapella, 
Campanus. 12. nQonintovaai b, corr. m. 2. nQog trjv tov 
itvtiXov nsQnpiQSiav] om. Proclus, Taurus, Hero, Sextus Emp., 
pBellns, Capella, Boetius; habent codd. (in b erasa sunt), Phi- 
loponus, Campanus. 13. slah] PF, siai uulgo. 19. iativ 
PF. 20. r^] om. B. %ai] tsnaiB, vnoXafipavofiivrjg B. 



6 STOIXEIiiK a. 

ttVT^g nsQt^eQBias. xbvtqov di roi; ijfUxvxHov t 
avTo, b xkI ToiJ xvxXov iffrtV. 

[&■'. I^x^iiuTtt BV&vyQajiita iSTi t« vtio tv&em 
nagiBxofiEVtt, T^CjiKcvQtt ^lv t« vno t^iwv, Ttrffi 
5 xlsvQtt Si Ttt vxo Tfffffaprai/, TtoXvjcXBVQa di ra wji 
nkiiovcav ri rtSGccQmv Bv&simv ^EQtixofiBva. 

x'. TiBv Si TQiJtXevQav ext^^iiTiav fsonXfvQov fik 
TQiyavov isTi to t«s rqets teas S^ov itXevQag, ttfo 
BxfXig Si TO ras Sva (lovag l'Cag ijjoi/ itXiVQug, exaXTjvi 

)0 di TO Tag Tffsig avCeovg l%ov TtXsvQag. 

xtt'. "Eti Se Tiav TQiitXBVQCiv Ux^^arav oQ&oya 
vtov [ilv TQCytavov ieri. ro sxov oq^^v yosviav, dj 
(iXvyfoviov Si ro e%ov afi^Xttttv yaviav, 6i,vy(avio 
Si t6 Tag TpEfg oisCag lj;ow yavCag. 

15 x^'. Tav Si TetQajcXsvQav ax-ri^aTav reTQiiyavo 
{liv iettv, o ieojtXevQov re iart xa\ oQ&oymviov, ei 
Qo^tixeg Ss, oQ&oyiaviov (isv, ovx iGonXsvQov i 
QOfi^og Si, 6 iaonXevQOV (liv, ovx oQ&oyiovtov i 
^ofi^osiSig Si To rag aTtsvavrCov :tXevQas ts xal y 

i!0 vCttg teag ttXX^^Xaig ^Z'"'? ° °''^^ ieo^XsvQov isriv 

19. Philop. in anal 11 foL 39'; cf. in Arifit. de anim. h 7. 
Boetiua p. 375,14 — 21. 20. Hero def. 43. 44. 45. Peellua 

1). 36. Boetiufl p. 370, 2. 21, Hero def. 4(i. 48. 47. Philop. 

iu anal. 11 fol, 39'. Paellus p. 37. Boetius p. 376, e. 22. Pael- 
lua p. 37. Martinmis Capella VI, 712. BoetiuB p, 376,14. ^d;i- 
^oe Qaleims XVin' p. 466. 

1. avTijs] avTOv B. TiegKpfseius] tov kvkXov Kpji(p£- 

eelas PBFV, sed xov iivkXov om, bp, Proclua, Hero, Capella, 
Boetius. yiivxgov Si — 2. iativ es Proclo p. IGO addidit 

Anguat eiecta defiiiitione III, 6, qaam omnes codd, boc quoque 
loco aic praebent; Tfi^fio KiiKllar iorl ta ncQic%6nevov ffl^fiw 
VTTO I* tv^eiag kkI itviiXov jtigiipeQeiaq ij iirClovoi ^ il^riovoe 
■qiiivvnXiov {xvy.lov hti om. rp; pro piiore ^ in BFV est Tjtai: 
iidaoovos P). eaudem babet Campanus; coutra Capella 



J 



EliEMENTORUM UBEB I. 7 

diametro et arca ab ea absciso comprehenditur. cen- 
tnun uero semicirculi idem est^ quod ipsius est circuli. 

XIX. Figorae rectilineae sunt^ quae rectis lineis 
comprehenduntury trilaterae quae tribuS; quadrilaterae 
qnae quattuor, moltilaterae quae plus quam quattuor 
rectis comprehenduntur. 

XX. Ex figuris autem trilateris aequilaterus tri- 
angulus esty qui tria latera sua aequalia habet; aequi- 
crurius uero, qui duo sola aequalia habet, scalenus 
autem, qui tria latera sua inaequalia habet. 

XXI. Praeterea uero ex figuris trilateris rectaugulus 
triangulus est^ qui rectum angulum habet; obtusiangulus, 
qui obtusum habet, acutiangulus autem, qui tres an- 
gulos suos acutos habet. 

XXn. Ex quadrilateris autem figuris quadratum 
est; quod simul aequilaterum est et rectangulum^ parte 
altera longius est, quod rectangulum est neque uero 
aequilaterum, rhombus autem, quod aequilaterum est 
neque uero rectangulum, rhomboides autem, quod la- 
tera simul et angulos inter se opposita aequalia habet, 
sed neque aequilaterum est neque rectangulum; re- 

Boetius et hanc et Procli omittunt; de Herone non liquet (Stu- 
dien p. 192). 8. axrjiiattt Bv&vygafifia] Pbp, Proolus; sv- 

9^yQ. a%. uulgo {sv^sfygafifia cp). iativ PF. Def. 19 

nu%o in 4 diuiditur; V hinc numeros om. 3. svd^stmv ygafi- 
ficav Proclus, Boetius. 6. tsttdQ(ov B. sv9'Si^v'\ nXsvgaiv 
Proclus, Boetius. 8. iativ PF. 9. raff 8vo\ Svo b, Pro- 

clus. yMvov Proclus. 10. nXsvQag] om. Proclus. Def. 20 
uulgo in 3 diuiditur. 11. 8s] P, Proclus; om. b; ts uulgo. 

12. iativ PF. n,Cav i%ov V mg. m. 1?, Proclus, Psellus. 
13. ykiav i%ov Proclus, Psellus; ytavCav (iCav V mg. m. 1? 
to i%ov — 14. Ss mg. B eadem man. okiymvtov tp. 16. o 

iativ laonXsvQov ts kuC Proclus. iatiVj o laonXsvQov ts om. qp. 

sTSQOfiTiitsg bis «p. 17. o] to Proclus. 20. o] om. Fbp. 

avts] ovts Si Fbp. iativ] om. Proclus. 



8 ETOIXEL<iM a'. 

oijts og&oymviov' ra 61 napa tavza ifTQtt7ilsv{ 

xy . TIaQtti.krjXoi bi6lv ev&elat, atrtVES iv t 

avT^ ixiTtide) ovSai xal ix^ak^ofitvai iig untiQov k 

6 ixdriQa tk j*£p*j fVl [irjddTSQa avjinixroveiv aXX^^Xat 

Airrj^ata. 
tt'. 'HiTtja&{a KMO aavTo^ arjfiiiov ial xav arjfiilo 
tv&(i^av y^a(itii}v ayayeiv. 

j5'. Kal jtex£(faaftivtjv sv&atav «atit tb avvixl 
10 in' tv^tiag ix^ttKttv. 

y'. Kttl Jittvtl xivT^a xkI SiaOT^[iaTi xvxkov ypo 
tfiEa&ai. 

S'. Kai naaas rag op&«s ycaviag iaas aXX^^loi 
Eivai. 
16 e'. Kttl ittv eCg Svo Evd^Etas tvd^ita ifiniTtroviti 
Tttg ivtog xtd ial ta avra iiiQf} ytnvias Svo OQd-m^^ 
iXaeaovtts noirj, ix^ttlko^ivas ras Svo si&tias A 
axsigov avjiniaTEiv, i<p' a firpjj sielv aC twv dvo 6{ 
&av iXaaeovss. 

23. Hero def. 71. PMIoponnB ia anal. II fol. 18'. Pietli 
p. 36. Mattianos Cap«I1a VI, TIS. BoetiuB p. 376, 28. ai_ 

1 — 5. Martianas Capella VI, 722. Boetius p. 877,4. AspBfdoB 
apud Simplicium in Ariat, de coelo fol. 149! tu «ttie ult^- 
fioTfe. 1. Philop. in ana.!. II fol. 9^, 10. 29. 2. Simplicins 

m phya. fol. 119. 3. Philop. in anal. II fol. 10.29. 4. Id. 
ibid. fol. 10. 6. Id. ib. fol. 10. 29. Ptodue p. 364, 14. 

1. TtTeaymva B. 2. tfane^eui h. Def 21 nulgo in 
def. Sa in 5 diuidnnt. 3. aagaXlTjXoi, 3c B. iv^itai tlm^ 
ProcIaB, PselluB. 4. is V. 5. avaxixxciv F. aU^kam 

om. F. 6. alt^iiata r^vic V, a^. iati revie BF, b m. 3. 
NumetoB om. F. 9. ix' cv9tiat «ntii lo avvtxis PBFbpj 



EliEIIENTORUM LIBEB I. 9 

Kqoa antem praeter haec qoadrilatera trapezia ad- 
pdlentur. 

XXm. Parallelae sant lineae^ qnae in eodem 
plano pofldtae et in utramque partem productae in in- 
Imitam in neutra parte concurrunt 

Postulata. 

L Postuletur^ ut a quouis puncto ad quoduis punc- 
tom recta linea ducatur. 

n. Et ut recta linea terminata in directum edu- 
eatur in continuum. 

m. £t ut quouis centro radioque circulus descri- 
batur. 

IV. Et omnes rectos angulos inter se aequales 
esse. 

Y. Ety si in duas lineas rectas recta incidens an- 
gulos interiores et ad eandem partem duobus rectis 
minores effecerit^ rectas illas in infinitum productas 
concurrere ad eandem partem^ in qua sint auguli 
duobus rectis minores. 



receptmn ordinem tuentur V, Proclus, Simplicios, Capellu, 
Boetios, Campanos. 10. inpdXXsiv V. 11. yQdtpsa&at] 

codd. omnes et Philoponus; yqdtpui ex Proclo recepit Augnst. 
13. aXXriXui,^'] om. V. 15. svd^sid tig P. 17. iXdrxovas 

Proclns p. 191,18 (non p. 364). tag Svo] PBVbp, dvo om. 
F, Proclus bis, Martianus Capella, Boetius, fort. recte. 18. 
oviAn^Tnstv tdg svd^slas iK§aXXofi,ivag itp' Proclus p. 864. <jvfi- 
nifnsiv dXXi^Xatg PY {dXXjjXaig corr. ex dXXrjXag P). 19. 

iXdooovsg^l Pp, Proclus p. 364; iXdttovsg uulgo. Dein add. yoo- 
viai FBVb, Philoponus; om. Proclus bis et Pp. In ed. Ba- 
8il. et apud Gregorium att. 4 — 5 inter communes notiones 
(10 — 11) leguntur {ndaai at og^al yoiviai taai . , slaC\ i^PaX- 
XoiiBvai at , . sv^siai , . aviinsaovvtai). Post ah. 5 in PF et 
V m. 2 et apud Campanum sequitur,: xal dvo sv^siag %mQiov 
fii) nsqii%siv. 



STOIXEIiiN a. 
Koival ivvotai.. 

Ta ta avra Isa xal RAAij^oig eerlv laa. 

Kal iav fffot? fffa 'XqoGrE^fj, ta ola iariv fsa 

Kcd eav atto {<Sav leK atpaiQsQ^., za xaralsi 

iexLV rea. 
[d'. Kal kav avCaois '<?« Jipo^rs*^, tk oAce lerl 
iiea. 

i'. Kftl ra Toil uvtov Sinkaeia iau kAA^Aoi? ierit 
?'. Kal ra rov hutou Tjfiiet] rea d^l^^Xoig ierlv. 
£'. Kal Ta icpapno^ovra eji aXl^qla fffa dXK->ilmis ieril 
jj', Kal ro oAov roO ft/pous uel^ov [ieiiv]. 
[■&'. Kal dvo {v9Etai ^[(optoi' ov nEQiixovOiv.'] 



'Enl rijg do&iiotjg av^sCas nfZiQaOfiiviig 
15 rffiycavov leoaXtvQov aver^^aae&ai. 

"Earta 1] So&Btea sv&Eia %c7tfQae[iEvri rj AB. 
^Et Si] inl tfjg AB Ev&eia; TQiymvov {eojtXevQOV 
0ver^aa69ai. 

KivTQip ^lv T£o j4 diaerijfiari Se ra AB nvxkos 

Kaiv. hv. 1 — 3. Martiamis Capella VI, 723. l. Phiiop. 

in anal. II fgl. 5. Boetiua p. 378, 1. S. Bootiaa p. 378, 5. 

3, Philop. 1. c. Boetins p. 378, 3. 4. Eutooina in Archim. 
nr p. 264, 27. 7. Philop. in anal. II fol. 5. BoetiuB p, 378,7. 

prop. I. Alejander Aphrod. in anal. I fol. 8', in top. p.ll. 
ThemiBtius phja. paraphi. fol. 35'. SimpUoiuB in phya. fol. 119, 
'Proclus p. 102, 14. 223, 23, Philop. in anal. II fol. 4'. Mart^ 
nus Capella TI, 724. BoetiaB p. 380, 2 [p. 390, G^2B]. ProcM 
p, 208—10 liberiiia pvopoait. repetit totem. 

1. diiai^ata ProcIUH p, 193. *oiv. ^vv. a.m BFV. 
ros om. PBF. 3. ("du iaoiq Proclua. iea lativ Proolns. ^ 

4, anh rncov faa] (aarv Proclus. 5. taa lazlv Proclus. 
ait. 4 ex comioeiitario Pappi irrepaisae uidetur; u. Procli)| 



ELEMENTORUM LIBER I. U 

Gommunes animi conceptiones. 

I. Quae eidem aequalia sunt, etiam inter se ae- 
qualia sunt. 

n. Et; si aequalibus aequalia adduntur, tota ae- 
qualia sunt. 

m. Et^ si ab aequalibus aequalia subtrahuntur, 
reliqua sunt aequalia. 

YII. Et quae inter se congruunt; aequalia sunt. 

VIII. Et totum parte maius est. 

I. 

In data recta terminata triangulum aequilaterum 
construere. 

Sit data recta terminata j4B, oportet igitur in 
recta j4B terminata triangulum aequilaterum con- 
struere. 

centro j4 et radio j4B circulus describatur BFJy 



p. 197,6 sq.; in omnibus codicibus legitur; quare iam aiite The- 
onem receptum erat (P); om. Martianus Capella et Boetius. 
Ante atx, 5 uulgo in codd. et edd. legitur: xal iav ano avi- 
anav taa dtpaiQsG^i, td Xoind ioTtv aviacc; om. 6, mg. Fb, in 
ras. postea additum p; nojti agnoscunt Proclus (cfr. p. 198,3), 
Capella, Boetius. ah. 6 — 6 reiicit Proclus p. 196,25, om. 
Capella et Boetius. «rr. 7 — 8 permutat Proclus p. 193, qui 
ea diserte contra Heronem sola air. 1—3 agnoscentem Euclidi 
uindicat p. 196, 17; om. Capella; «rr. 8 etiam Boetius om. 
aMT. 9 om. Capella, Boetius, Proclus, qui diserte id improbat 
p. 184,8. 196,23. Hoc loco habent Vbp; cfr. Philop. ad phys. 
fol. 10; xal dvo svd^siag x<>iQtov fiij nsQiixBi^v B; de ceteris u. 
ad p.8,19. 8. iativ] PF, iati uulgo; comp. b; item lin. 9. 10. 
10. in* ttXXrjXa] om. Proclus. iativ] slaC B. 11. iatCv] 
om. Proclus; comp. b; //at F, slvai P. 17. sv^sCag] om. 

BFbp. sv^sCag nsnsQaafisvrig P. 19. fiiv] om. bp. xat 

diaatrjiiati Bp. di om. BFbp. 



12 



rroixEiKN 



Ytygaqi&G} o BF^, xal itaf.iv xdvTQp (tiv ta B Stcf 
fftTJfiRTt tf^ Tf5 SA xvxkoq y£ypa<p9-(o 6 AFE, 
ano ToiJ r OijfiBiov, xa&' o rd(ivov6Lv aXXt^lovg t 
xvxXoi , inl T« A, B ariftcia imisvx^<»9av fvQitc 

6 FA, rs. 

Kal ^xil To A erjftitov xtVrpoi' iexl zov F^S 
xvxXov, Caij iarlv ^ AF rjj AB' ndXiv, ^nel 
arjfitrov XEVTpov iotl Tov FAE xvxXov, itsri iorlv i 
Sr rp BA. iSeix^t] dh xal ij FA t^ AB ioij- 
10 Tsga apa Tcav TA, FB t^ AB icxtv ffljj. ra di vm 
KUTto t<sa xal alk^^Xots iatlv l'ea " xai tj VA apa i 
FB ionv 1671' "^ ^pits «p« at FA, AB, BF i'aat 
i^Aatg EifftV. 

CaoaltvQov «p« ^flri to ABF T(/tyiavov. xttl avvi 
16 ioTKtai iicl T^e do^fi^tfjjff e^dftas jifiripaO^cvJjs t^^sAB. 
['£3it T^g do&siajjg cipa tv&sias JtsitEQaaiisvTig Tpt- 
ycovov iao^isvQov avviOTttTai\ oniQ idsi. noiijaai.. 



npos Tm So^svTt ariiisia tij So&siati sv&sia 
20 Cativ £v96i:av &ia&ai. 

"EsTta To ftiv Ho&iv arifisiov to A, ^ Si 6o&et0 
tv&sltt 7} BF' dtt 5ri wpos iro ^ aTjfitip rjj So&siO^ 
tv^Eia T^ BF tarjv sv&etav &i69ai. 

'Entlsvx^a ya^ azb toij A aijfisiov inl t6 B tfiS 

86 fulov ti&ettt r) AB, xal avvtOraTCj in" «ur^s rpty 

vov laonXsvgov ro ^AB, x«l ix^spXrjad-aaav . 

II. ArchimedeB 1 p. 14,1. Boetina p. 880, 3 [p. 391]. 



uw] 



1. £rj] P,Vm.li ndS Fbp.Vecorr.; PB^ in rM.B,.| 
t£] td qj. B. ^rE] P, Vm. 1( PAE BFbrf 
i. PoBt A tM. 10 litt, b. /flti'»' P. r<dSj .(J I 



ELEBCENTORUM LIBER I. 



13 




et rursus centro B radio 
autem BA circulus descri- 
batur AFEf et a puncto 
r*, in quo circuli inter 
se secant; ad puncta A, B 
ducantur rectae rA. FB. 
iam quoniam punctum 
A centrum est circuliFzf ^, 
erit AF = AB. rursus quoniam B punctum centrum 
est circuli FAEy est BF^BA. sed demoustratum 
est etiam FA »> AB. quare utraque FAy FB rectae 
AB aequalis esi quae autem eidem aequalia sunt, 
etiam inter se aequalia sunt [x. ivv. 1]. itaque etiam 
FA »= FB. itaque FAj AB, BF aequales sunt. quare 
triangulus ABF aequilaterus est; et in data recta ter- 
minata AB constructus est. quod oportebat fieri. 

II. 

Ad datum punctum datae rectae aequalem rectam 
constituere. 

Sit datum punctum A, data autem reeta B F. opor- 
tet igitur ad punctum A datae rectae BF aequalem 
rectam eonstituere. 

ducatur enim a puncto A eid B punctum recta AB 
[atr. l]y et in ea construatur triangulus aequilaterus 
^AB [prop. I], et producantur in directum rectae 

ras. eet in V, JB in B; BTJ P. 7. iativ Cari BF. 8. iativ 
P. FAE} in ras. B, ATE P. 12. i^ari iaxiv V. AB] FB 
cp. 14. iatCv P. avviaxatai PBV (in b non liquet). 16. 
ifflT^g — 17. avy£(rtatai om. codd. omnes ; e Proclo solo p. 210 
recepit An^t; oiz genuina sunt. 22. t^ So^siajj €v^eia]F; 
om. Theon (BFVpb). 23. BT ev^eia V. 24. yorf' om. 

F. 26. dAB] eras. F. Ante i%§epX. in V add. supra: nQoa-. 



Bv&Bias ratg ^dA, jdB cvQitai al AE, BZ, xal xivzpp 
fiiv tp B diaOT^fittTi Se T^ BF xvxAos yByp«g>&o} 6 
rH&, xal ntiX.iv xivzQa za d xwi dtKffTijfittri tw ^H 
xvxXoi; ysy^Kfp&ta 6 HKJ. 
5 'EnBL olv To B etiiittov xiVTffov ^CtI iou rH&, 
i'6rj iffilv ij BP Ty BH. itaktv, iasl to ^ fftjjifrov 
xivTQov iCTL roO HKA xiixkov, Heii ioTlv tj ^ /i ttj 
jdH, tov 7] ^A T^ /iB terj iOTiv. Xomij Sqr ij AA 
AoiTtij Tr} BH iaxiv i!f}<q. iSBt-f&rj 6s xal ij BT 
10 zri BH /'fftj- txKTEpa aga rav AA, BF xy BH iezt 
['erj. TK 6h za avza tea xal ^AiijAoij,- ioTlv l'(Sa 

jj AA ttQtt Tjj Br EffTtv rotj. 

Jlpog aQCt Tp do&BVTt erjftBia Ta A zrj iJo#ttffjj 

tv&tia zf] BF terj iv&aitt XEiTai ^ AA- owfp ISbi 

15 noiijaat. 






z/iio do9netav Bv&Btmv avioav ano 
fiBi^ovos Ty iXaeeovi Cerjv tvd-Btav dipsXBti 

"Eazmaav «l So&steai Svo sv&Btai &vtaot al ABA 

2u r, mv ^Bi%mv Ibtm ^ AB- Set Sij axb T^g ftEi^ovoa 

Trjs AB Tfj iXdeeovi z^ F Catjv sv&Btav aipsXstv. 

Ksie&ca Jipog Tto A arjiisiej zjj F sv&sia ia^ ijl 
AA' xal xBVTQip (iBv Tra A Siaaztjftazt 8s za AAfM 
xvxlos ysyQttip9a 6 ^EZ. 

m, Boetiua p- 380,5 [p. 302]. 

1. erfff/os FV. 3. xfvxQa (liv V. kb] bifi B (in 
et initio linn.). nal SmoTqiiaxi} SLaet^fian Se V. 5. rH6 
xyKiotJ BFV, P m. chc. 6. BT] TB F. «ol naliv V; 

iialiv di (aupra) p. 7. iuTtv P. 8, iori'i'] PF; iati uulgo. 
9. Tj] om. b. 10- Ts Bfl-] (itlt.) Buprab. 11. tiiti](M.) 
-a in ras. P. 12. BTJ TB F, 13, Ant« neoe ras. uniaa 

Utt, b. 18. ilaTTOVt BF. tutfeiuv] om.Proclus. 19. *i!o] 
oiD. F. ttviaoi] dv- snpra m. 1 F. 20. Post V rae. 1 litt. 



£L£M£NTOBUM LIBER I. 



15 




dAy AB, ut fiant AE, BZ, et centro B radio autem 
Br circulus describatur [afr. 2] FHS, et rursus cen- 
iro ^ radio autem AH circulus describatur HKA. 
iam quoniam B punctum centrum est circuU FHe, 

erit Br= BH. rursus quo- 
niam A punctum centrum 
est circuli HKA, erit 
AA = AH, 
quarum partes AA, AB ae- 
quales. itaque AA = BH[x. 
ivv.d], sed demonstratum est 
Br=BH. itaque utraque 
*^ AA, BF rectae BH aequalis 

est. uerum quae eidem aequalia sunt, etiam inter se 
aequalia sunt [x. ivv. 1]. ergo etiam AA^BF. 

Ergo ad datum punctum A datae rectae BF ae- 
qualis constituta est recta AA) quod oportebat fieri. 

m. 

Datis duabus rectis inaequalibus rectam minori ae- 
qualem a maiore abscindere. 

Sint duae datae rectae inaequales AB, F, quarum 

maior sit AB. oportet igitur a 
maiore AB minori F aequalem 
rectam abscindere. constituatur 
ad A punctum rectae F aequa- 
lis AA [propr. II], et centro >^ 
radio autem AA describatur 
circulus AEZ [aCr. 2]. 

P, ut lin. 21. 22. 22. Post %bCc^o} in P supra scr. m. 1 yap, 
idem V mg. 23. AdVi^i.) in ras. V; utrumque corr. ex AE 
P m. rec. 24. z/EZJ ex EZ/ P m. rec; ZE^ B. 




16 ZTOIXEIAH a'. ■ 

Kal iasl to j4 eii^tiov xdvtpov iatl tov ^EZ 
xvxKov, fffjj ietlv 1] AE rij A^' aXkk xai ^ T ttj 
A^ ietiv Harj, ixardpa apa rav AE, r zy AA iariv 
tet}' Sers xkI 17 AE t^ f ieriv iat], 
5 ^vo aga do&8iemv sv&tiav avCetov xav AB, F 
aao T^g fiei^ovog i^s AB TJj ikaaaovi x^ P larj aipjj- 
pjjrat f] AE' OTtsQ HSei. noi^eai. 



'Eav dvo Tpiyava rag 3vo nlsvpag [ratg] dvsl 1 

10 mkivpatg Haag Ijt; ixatiQav ixatipa xtti zijv ] 
yiavCttv ffi yatvia IOtiv exv ''^V^ ^^^ ^'^" /"Oav 
av&eiav «SQisxofiivriv, xai tijv ^aaiv tj; fiaasi 
l'a7]v e^et, xal ro tpiyiavov ta tptyavp taov 
iatai, xal ai XotJtal yuoviat tatg iotaatg yta- 

15 viaig taat iaovtai ixati^a ixatiga, v<p' ag al 
leat n^fvpai vaotsivovatv. 

"Eatia dvo xqiyava ta ABF, ^JEZ tag Svo xltv- 
gag tag AB, AT tatg Sval TtlsvQatg taig jdE, ^Z 
leag iiovta ixatipav ixaripa tijv {ilv AB tfj AE 

20 tijv Si AV tfj AZ xal yoviav tijv vno BAV ytavia 
tfi vnb E^Z teriv. Xiyia, ort xal ^aetg t} BF ^aeet 
Tfj EZ teri iativ, xai to JBF tQiymvov ra z/EZ 
tgtycova tOov ietat, xal a/ Xoiital yaviat tatg Aoi- 
xtttg ytaviaig /'ffat eaovzai ixaiiqa ixatiga, v<p' ag 

25 aC teai, 7tf.tvQa\ vxotEtvovaiv, rj /ilv urco ABF t^ 
vjto JEZ, r] Si vno AFB Tfj v«o AZE. 

'E^atfjto^Ofiivov yap tov ABF tQiyavov inl ri> 



IV, Sohol- in Papptun UI p. 1183, 32. Boetana p. J 



— T. Mnltas litt. fig^ in raa. P m. tec., nt supn. 1. ^] 



ELEMENTOBUM LIBER I. 



17 



£t quoniam punctum A centrum est circuli ^EZ^ 
est AE^ A/l\ uerum etiam r= AA. itaque utraque 
AEj r rectae AA aequalis est; ergo etiam AE = P. 

Ergo datis duabus rectis inaequalibus AB, F a 

maiore AB minori F aequalis abscisa est AE] quod 

oportebat fieri. 

IV. 

Si duo trianguli duo latera duobus lateribus alte- 
rum alteri aequalia habent et angulos rectis aequali- 
bus comprehensos aequales^ etiam basim basi aequa- 
lem habebunt; et triangulus triangulo aequalis erit, 
et reliqui anguli reliquis aequales alter alteri, ii sci- 
licet; sub quibus aequalia latera subtendunt. 

Sint duo trianguli ABF, ^EZ duo latera AB, 
jl AF duobus lateribus AE, AZ aequa- 
lia habentes alterum alteri; 

AB = AE et AT = AZ, 

et L BAr= EAZ, dico, etiam esse 
Br=EZ et AABr= AEZ, et 
reliquos angulos reliquis, alterum 
alteri, aequales, sub quibus aequalia 
latera subtendant, L ABF = AEZ 
et ATB = AZE. 
Nam si triangulum ABF triangulo AEZ adpli- 




dertum m. 1 b. Q. AB] B supra scriptum m. 1 b. 9. tais\ 
om. Pp; supra b. 10. ixsi (scr. ixv) ^* ^^^ ytoviav ymvia 

lariv Proclus, xriv fii'a» ycaviav rj fiia yoov^aBF. 12. cv^stcov^ 
TtXhVQOiV Proclus. 15. SHtttSQa inatsQa] om. Proclus. vq>*\ 



i(p' b. af] om. V. 
Hai] comp. supra F. 
E^Z] E^.eras. F. 
supra itp'. 

Euclides, edd. Heiborg et Menge. 



18. dvai V. 19. ixovti 9. 20. 

BAT] ASr F, sed AB eras. 21. 

22. iatiY. 24. vtp'] sic b m. 1, sed 



18 ETOIXEIiiS o'. 

^ET, tiflyfovov lud xiifttiivov tov ftiv A 
ixl t6 A aijfutov t^s Ae AB tv&iia^ ix\ xijv ^E, 
iif^tiQitoen xfd t6 B ffijfutov im to £ dia ro Istjv sivat 
jfir AB TQ AE- itpa^iioodstig dii t^s AE imi z^v 
t> ^E i^ioifjtoeet xal ^ AF ii-Qela ixl tr,v ^Z dia to 
ftftjc flvtti Tijv vjio BAF yaviav ttj irxo EAZ' mtf» 
xul ro r Oijutiov tTll ro Z otjfifiov i<pa$fioSn, Sict 
lo ftfiji/ Tcd?.iv ilvui njv AF tfi AZ. «AA« fi^w xaX 
th B iai tu E iiptjpfioxtf mort ^aei^ i} BF inl ^- 

10 fftv i^v EZ itpa^fioeti. ti yicQ loir ftiv B ixl ri 
iqMifHOBavtQS roi) di F inl zo Z ri BF ^aot^ ial 
EZ ovx i^agnoett, dvo ev^stai x^Qtov xtgtiioiiSiV' 
CxtQ iailv u&vvttxov. i^apfioeti a^a ri BP jkt<fc; ial 
rijv EZ xal ieii avry effiai- aett xal oiov to ABV 

15 TffCyavov inl oAov to jdEZ Tptytofov i<pa(f(i66H xol 

i'6ov uvTa iotai, xal ai kotxal yaviai ixi r«s' i.atKas 

yavias ipagnoeovai xal taai avtats iaovrat, ri fiev 

wro ABr Tfi izo JEZ ^ di vno AFB tij ioto AZE, 

'Eav apa dvo tgiyava tus 6vo idfvpag Iffs) ^vo 

2ii nlivgats iaag IxV txatigav ixattQtt xal t»)i/ ymviav 
ty yavla tariv ix^jj tijv imo Troi' fBrav £v9ttiov nigttx* 
(livjjv, xal Ttjv fiaaiv rjj ^uati tyijv i\tt. 
yavov ta tgiydva tsov iarai, xul uC loinal yiDvitu 
ttttg iloinais yavtuis ^aui iaovtai ixarigtt exatiffa, 

26 iiy' ag ttl iaat xXsvQal vaoTtivoveiv ' oxtQ idti Sit^ai. 






av 



L 



I. xfi>ini9itiivov V, aed npoa- pmictiB del, jis»] supra 
m, 1 F. 2. a] m ras. b. t^v] rj p. 4. 3^] FVbp; 

a^ PB; cfr. prop. 8. 6. BAT] poat rae. V; ABF B. 

fH^iZ] JEZ B. 8. fivut Koltv fi, 0. ^q](Ep;H>af( b. 
ioii'»] om. T. 16. tatt lomak yiavCaiq BF. 17. itpv^^ 
flocCT.* P. avrait] liUqiBis F. J9. Sva] (alt.) ^ F. 



Sk 



i 



ELEMENTOKUM LIBER I. 19 

caerimus et punctum >^ in ^ puncto posuerimus^ rec- 
tam autem AB in ^E, etiam B punctum in E ca- 
det; quia j^B «= ^dE. adplicata iam jiB rectae ^E 
etiam j4r recta cum ^dZ congruet, quia LBAF^ 
EAZ. quare etiam punctum F in Z punctum cadet^ 
quia rursus Ar=^dZ. uerum etiam B in E ceci- 
derat; quare basis JSF in basim EZ cadet. nam^ cum 
B rsL E^ r uero in Z ceciderit, si ita basis BF cum 
EZ non congruet^ duae rectae spatium comprehen- 
dent; quod fieri non potest [x. iw. 9]. itaque^basis 
jBjT cum EZ congruet et aequalis ei erit [x. ivv. 7]. 
quare etiam totus triangulus ^^Fcum toto triangulo 
j^EZ congruet et ei aequalis erit^ et reliqui anguli 
cnm reliquis congruent et aequales iis erunt; L ABF 
= AEZ et L ^TB = AZE. 

Ergo si duo trianguli duo latera duobus lateribus 
alterum alteri aequalia habent et angulos rectis ae- 
qualibus comprehensos aequales^ etiam basim basi ae- 
qualem habebunt, et triangulus triangulo aequalis erit, 
et reliqui anguli reliquis aequales alter alteri, ii sci- 
licet, sub quibus aequalia latera subtendunt; quod erat 
demonstrandum. 



Taiq\ om. Pbp. 8vaC V; in p 8vo nlhVQaiq deleta sunt 

m. 1. 22. ?g€i Hariv BF. 26. vy'] corr. in /9' m. 1 b. 
vq>' Sg — vTTOTSLvovaiv] mg. m. 1 P. 



ETOTXEmN 



Tav ieoGxilmv TQiyavav al hqos ijj ^aoet 
yavlai taai cAAijAKts eleiv, xal XQoGExfikri&ei- 
Biov twv ("atav fvQEidiv al vjio rijv ^aatv yto- 
5 viai Caai all-^laig laovrai. 

"Eata tgiyiavov iaoaxeies ro j^BF Caijv ejoj' rijr 
^B itlevQav tij AT nkevpa, xai npoaEx^e^ktja^aaav 
irc' ev^eiag zaig AB, AT ev&etai al BJ, FE- i/yro, 
oti ij fih' vna ABF ytnvia rij vTth AFB tarj iStiVf . 

10 17 6h vito rS^ tij vjtc BFE. ] 

slKrjfp^G} yag i%l r^s BA Tv;i;ov a^^etov ro Z, 
Xttl ttfptiQ^e&m «Ko T^s fif^fiovos T^g AE tj; Hdaaovi 
t^ AZ Hatj 1) AH, xal instavx&aaav al ZF, HB 
tv^elai.. 

15 ixeX ovv i'ai\ iatlv 17 (ilv AZ tjj AH 17 di AB 
TJ5 Ar, Svo Sij at ZA, AF Svol tais HA, AB laai 
e{a\v ixariQa axttziffa' xal ytoviav xoivijv TttQiiy^ovai. 
r^v VTto ZAH- ^decs aga ^ ZT ^aaet tji HB tOi} j 
iativ,Xttl ToAZr xQiyavov ta AHB tQiycSva laov ' 

20 ierai, y.al at koiaal yaviai r«rs Koiaatg ymviaig taai 
iaovtai ExatiQa exatiQa, vip' «g at teai nXevQal vaO' 
ttivovaiv, 1} fiiv vnb AFZ rjj vno ABH, ^ di vno 
AZr t{j irjto AHB. Xttl ixei oXri 7} AZ oAi; t^ AH 
iativ te-r), av ij AB ij; ^F ioziv tart, loniij UQa ij 

26 BZ Xomfi tfi rn iativ tet]. ideix^ Se xal 17 ZT 
ry HB fOij" Svo S^ at BZ, ZT Svel ralg FH, HB 

t. vfoe] tQO b, aed corr. m. 1. 3. aH-^laisi om. Pro- 

olns, sfafti] P, Proclna, comp. b; ttai nnlgo. 6. dU^lais] 
om. ProoluB. leovrai] elai ProcluB. 1. wljupoj xlevpav 

ip. 8. tv9e(at] eufre/we B, 9. ATB] ABF ¥. 10, 

TBJ fffij itzi p et V m. recentissima. 17. Tcefirxovoty 



ELEMEKTORUM LIBER L 21 

V. 

In triangulis aequicruriis anguK ad basim positi 
inter se aequales sunt; et productis rectis aequalibus 
anguli sub basi positi inter se aequales erunt. 

Sit triangulus aequicrurius -^-BFhabens jiB = Ar, 

et producantur jiBj AF ia directum, 
ut fiant B J, FE. dico, esse 

L ABr= ATB 
et L rBJ = BFE. 

Sumatur enim in BJ quoduis 
J[ punctum Z, et a maiore AE minori 
AZ aequalis abscindatur AH [prop. 
^ B ni], et ducantur ZF, HB rectae. 

iam quoniam AZ = AHet AB = AF, duae rectae 
ZAj AF duabus HA, AB aequales sunt altera alteri; 
et angulum communem comprehendunt ZAH. itaque 
Zr=HB et AAZr= AHB, et reliqui anguli re- 
liquis aequales erunt alter alteri, sub quibus aequalia 
latera subtendunt [prop. IV], L AFZ = ABH et L 
AZr = AHB. et quoniam AZ = AH, quarum par- 
tes ABj AF aequales, erit BZ = FH [x. Svv. 3]. sed 
demonstratum est etiam ZF = HB. itaque duae re- 
ctae BZy ZF duabus FH, HB aequales sunt altera 
alteri; et L BZF = FHB et basis eorum communis 




V. Simplicius in phys. fol. 14\ Boetius p. 380, 13 — 15, 
ubi sic fere scribendum : si triangulus aequalia latera habeat, 
qui ad eius basim anguli sunt, aequales alter alteri sunt, et 
aequalibus lineis [productis] et sub basi eius anguli aequa- 
les utrimque erunt. 

PVp. 19. ictiv] PF, comp. b; ^flrW uulgo. 25. Ante BZ 
ras. est unius litt. in V. 26. HB] Bif V, corr. m. 2. 

dv6i] e corr. V. 



22 ETOISEiaN c'. 

l0tti bMv iytiTeQtx ixat^QK' xal ymviu i^ vjto BZF 
yavCa tFj vno THB iOt], xal paais avztiav xoivii i) 
BF' xal TO BZr a^a XQiyavov rp PHB TQiyavca 
taov ^mai, xal aC loinal ymviai. tatg XoiTTutg ymvCaig 
5 Ajo!t ISovTai ExaTega maTiqa, vtp dg aC tSai xJ.Evpal 
vaoTsCvovetV fatj «p« datlv ti fiJv vjto ZBFr^ vno 
HFB i} Si vno BTZ t^ wito FBH. dm} ovv oA»; ■>) 
vno ABH yavia oXrj Tjj vnb ATX ytavCa i6et%&7i 
Carj, (ov ^ vno FBH ry vno BFZ tarj, iow^ aga ^ 
to vjto ABF loin^ t;} vnb ATB iativ fffij' xnC aCai 
jtQog zij jiaaii tov ABF TQLyavou. iStCx^T] 61 xal 
^ vxo ZBV TTj vno HFB tar\' xaC aiaiv vna triv 

Tcov apa CooOXiKav tQiytavcav al nffhs tfj ^aost^ 

15 yavCai taai akX-qkais tCaCv, xaX nQoasx^XijS-Biaav covl 

iemv sv&timv ai vno rijv ^aaiv yavCai. faat a^^ijAntjl 

isovrai ' oatf/ Idti dtt^ai. 



'Eav TQiyavov aC dvo yavCai l'aai (cAA^J.a 
) aaiv, xal a[ vno tag 'iaag ymvCag vnotBivovaat 
nf.EVQal laai. «AAijAais taovrai. 

"Eata tQiyavov to ABT fffijw ix"'" ''^V" '^^a ABF 
yaviav tij vno APB yeavCa ' kty<a, oti xal itXtv^a ij 
AB jtkevQa rfj AF iariv f<Jij. 
3 (i yitg aviOog ictiv rj AB tjJ AF, 7} ivifa avttov 
liEit<ov ierCv. ierm (isC^mv ^ AB, xal dip^jQ^^a&io dno ^ 
T% fisit,ovog r^g AB rij iXdrrovi rg AF tar] i] ASf,.^ 
jtrei indtsvi9io ^ z/r. i 



fi. iazlv op« V. ZBT] iu ras. V. 7. HrjBl oorr. ex 
HB V. 9. ftni] (alt.) i«t» ftrij V e corr. 10. ewo] (alt.) 



ELEMENTORUM LIBER I. 23 

BF. itaque etdam ABZr= FHB, et reliqui anguli 
reliquis aequales erunt alter alteri, sub quibus aequalia 
latera subtenduni itaque L ZBr= HTB et BFZ 
= FBH [prop. IV]. iam quoniam L ABH= AFZ, 
ut demonstratum est, quorum partes FBH, BFZ ae- 
qnales, erit LABr= AFB [x. iw. 3]. et sunt ad 
basim positi trianguli ABR uerum etiam demonstra- 
tum est LZBr= HFB'^ et sub basi sunt. 

Ergo in triangulis aequicruriis anguli ad basim 
positi inter se aequales sunt, et productis rectis ae- 
qualibus anguli sub basi positi inter se aequales erunt; 
quod erat demonstrandum. 

VI. 

Si in triangulo duo anguli inter se aequales sunt, 
etiam latera sub aequalibus angulis subtendentia inter 
se aequalia erunt. 

Sit triangulus >^Br habens LABr=ArB. dico, 
esse etiam AB = AF. 

Si enim AB rectae AF inaequa- 
lis est, alterutra earum maior est. sit 
AB maior, et a maiore AB minori 
AF aequalis abscindatur ^IB [prop. 
ni], et ducatur ^F. 

VL Boetius p. 380, 15. 

supra m. 1 B. tVi? iativ F; tari ieti B. daiv P. 11. 

ABT] AFB B. 12. ifTB] e corr. V. 15. daiv'] PF; 

comp. b; dai uolgo. ^ nQ0GB%6XriGQ'Biamv P. 19. dXXriXaii] 
om. Proclus. 20. maiv] Proclus, PF; mai uulgo. 0^] om. 
F. ^ 21. aXXijAoftg] om. Proclus. ^aovxaC] stai Proclua. 

26. ^ iziQu] fiia in ras. 6 litt. P m. recent., BxiQa p et b m. 1 
rt supra insertum). 27. iXdaaovt BFV. 




r 



24 rroiXEiflN u'. 

'Eitd ovv latj itSTlv 71 ^B ifj AF xoivi] dl fj BT, 
Svo Srj aC JB, BT 6vo rats AF, FB leai tlalv 
eWTBQU ixKTd^a, xal yiaviu ij vTto ^BV yavCt^ tij 
vjcb AFB iariv ta?}' ^aaii Kpff ^ z/r ^aSti. t^ AB 

5 itfjj idttv, xal to ^BF TQiyiovov rco AFB rpiyava 
Haov eeTtti, t6 iXaaeov ro ftftSorc o^bq utotcov ovx 
UQtt Kviaoz iariv 17 AB rjj AF' iaiq agu. 

Eav UQtt TQLycovov tti Svo ymvCai. Caat tti.?.rji,uig 
meiv, xal ttC ijro rag Cettg yavCas vaoTHvoveai nXcv- 

10 Qttl iattt akkjiXais ieovTttf oxe^ f'Sti Sst^ai, 



1 



i 



'EnX Ti\i «Ot^s ev&Eta^ Svo Tatg avrats 
i^v&ECttis uKXtti Svo Ev&iiai taat ixaTdpa ixa- 
TSQtt ov aveTtt&^^eovTttt npos aXXc} «al ctXX^ 

15 fftjftftG) iicl Ttt avTcc ^fpij Ttt avzK nigttzttm 
ixovaat Tttts ii, ttQXVS svQsiatg. | 

Ei yag Svvazov, ixl t^g «utijs sv&tias tijs AB 
Svo iKte avTttie Bv&tiais tafs AF, FB ttllai Svo 
tv&Biai at AA, AB teai ixtttiQa ixaTBQa avvEBTtt- 

20 Ttaaav Jtpos «A^c) xal KlXa erj(iBia Tip tb F xal A 
ixl Ttt ttVTtt fiBQTj za avza ^iQaza 'ixoveac, (bOt£ rojjv 
slvat tiji' y,\v PA t^ AA 16 «vio niQag tjjouflftv 
aitry to A, rijv Si FB rrj AB t6 «uto xbqus ^X"^" 
aav avTtj zb B, xal iat^evx&c^ ^ F^. , 

26 'EhbI ovv fffgj iaTlv 7} AF z^ AJ, fff»; iezl xal 

2. Svai T. 3. kk/] bis B (jn fine et init. linn.). 
Poat JBV raa. 3 litt, F. 4. .^rSl .^BT, sed B in ras. F. 

6. JBT] corr- eiJBT V; ASr b. .irB] torr. os.jrB 
V; in raa. B; JFB b. 6, ^'iaiiov B. 7. avtaos] anpra 

m, 3, in textu iuit>av m. rec. in ras. P. 9. mct»'] PF; mot 

unlgo. or] supra P. 12. SvaiY. Post tmfs raa. 6 litt. 
P, 14. ov aTa^^aoviai (acr. avilra9.) ixaziga iMarifa Pro- 



ELEMENTORUH LIBER L 25 

iam cum AE = AF^ et BF commmiis sit, duae 
rectae jdBj BF daabos AFy FB aequales sunt altera 
alteri, et L ^Br^AFB. itaque ^r=AB et A 
jdBr= AFB [prop. IV], minus maiori; quod absur- 
dum est [x. iw, 8]. itaque AB rectae AF inaequalis 
non est; aequalis igitur. 

Ergo si in triangulo duo anguli inter se aequales 
sunt, etiam latera sub aequalibus angulis subtenden- 
tia inter se aequaUa erunt; quod erat demonstrandum. 

VII. 
In eadem recta iisdem duabus rectis aliae duae 
rectae aequales altera alteri non constituentur ad aliud 
atqne aliud punctum ad eandem partem eosdem ter- 
minos, quos priores rectae, babentes. 

Nam si fieri potest, in eadem recta AB duabus 
iisdem rectis AFj FB aliae duae rectae AA^ AB ae- 

quales altera alteri constituan- 
tur ad aliud atque aliud punctum 
r* et z/ ad eandem partem eos- 
dem terminos habentes, ita ut 
FA = ^ Ay quacum terminum 
habet communem A, et r!B=z/ J5, 
quacum terminum habet communem B, et ducatur FA, 
lam quoniam AF = AA, etiam i ATA = AAT 

Vn. Boetius p. 380, 19. 

clns. 19. af] om. P. fivvh<ixdztQfiav\ corr. ex crvvcffToxray 
B. 21. Post ^i^Ti add. ra R z/ P m. rec, mg. m. 2 FVp. 

Post l%w)cai in P m. rec, Vp m. 2 add. ra A, B; in FB 
add. xaCg i^ ciffXVS Bv^siaig ; in F praeterea m. 2 : ^roi ta A^ B 
(post sv^tUii). 22. JA'] AJ BF. 24. rd\ z/F BF. 

26. ftnj] pofitea add. P, Post A V add. hv^sia P m. rec. 

ifixiv P. 




26 STOIXEIiiN u. ■ 

yavitt t] vao AFjJ r^ vtco AJF' (tei^tav apa r/ vxo 
ji^i^r f^s vjto iJrB- XoH^ ttQa 7] vjtb /V/-B \iti%tav 
ifftl r^s vno dFB. naXtv imd iinj tezlv ij FB rjj 
iiS, fffjj igrX Kttl yeavia rj vito F^B ymvia tfj vno 
5 ^TB. iSeix^i] di avTijs xal noXXa fiEiiav oxeQ 
ifftli' d6in'ttrov. 

OvK apor ial rtjg awr^g evQEiag dvo ratg avtais 
iv&fiaig (ilXai dvo sv&etai teai sxatBptt ixttrdga 
iSvata&^aovrai rtpog alXp xal aXXo) atifttiat hxl ra 
10 avta (liqr] ra avTa niQata ^xoyff«t Tttiq i| ^&X^S^ 
£V&Biaie' oasQ iSgi 6st^ai. M 



'Eav dvo TQiyava r«g bvo Ttlsvgag [tais] Svo 
itlevQttts COttg Exji ixatEQttv ixtttB^a, J^XB ^^ 

3S xal rijv fiaaiv r^ ^kOei tOrjv, xal t^v ytaviav 
r^ yatvia teijv ^^si tijv V7t6 tiBv teav sv&si^v 
nBQisxo(iivy\v. . 

"Earto Svo TQiyavtt ra ABF, ^EZ rK^ Svo aXsv- 
Qttg tagAB, AP tatg Sva xi.EVQats raFg jdE, ^Z fffag 

20 i%ovra ixaripav ixariQa, rijv (liv AB ry ^B ri^v 
Sb AV r^ ^Z- iiira Se xttl paoiv rriv BT ^aHsi tfi 
£Z fffijv kiyca, ort xal yatvia r} vno BAF ytavia tfi 
vTto EAZ istiv ffffj, 

'E^aQ(i.oi,o{iBvov yap tou ABF r^iytovov izl x^fl 

25 AEZ tpiymvov xal rid-Bfiivov rou (liv B arniBiov h^M 
to E 0i][iBtov T^ff Se BF tv&sittg inl t^v EZ «Viic^l 
fioOft xai tb r Srjfistov inl ro Z Sia to tatjv slvULi 
Ttjv BF Tij EZ- itpaQfioeaeris ^V ^V? ^^ ^^*' ^V'" BZM 

a. zfis] corr. n rfi P. B. TB] e oorr. V; BTBF. 
toitV P. r.tJB] B.dr p. 5, dFB] BTJ p. 13. 



ELEMENTORUM LIBER L 27 

[prop. VJ. quare L^^r> jdFB [x. iw. 8]. itaque 
multo magis L r^B> dFB [id.]. rarsus quoniam 
FB = jdBj erit L FJB = JTB [prop. V]. sed de- 
monstratum est, eundem multo maiorem esse; quod 
fieri non potest. 

Ergo in eadem recta iisdem duabus rectis aliae 
duae rectae aequales altera alteri non constituentur 
ad aliud atque aliud punctum ad eandem partem eos- 
dem terminos, quos priores rectae^ habentes; quod erat 
demonstrandum. 

vin. 

Si duo trianguli duo latera duobus lateribus ae- 
qualia habent alterum alteri et praeterea basim basi 
aequalem habent, etiam angulos aequalibus rectis com- 
prehensos aequales habebuni 

Sintduo trianguli^Br,^JBZ 
duo latera ABy AF duobus la- 
teribus JE, z/Z aequalia ha- 
bentes alterum alteri, 

AB = ^E et Ar = dZ, 
et praeterea habeant BF = EZ. 
dico, etiam esse L BAF = EjdZ. 

nam triangulo ABF ad triangulum AEZ adplicato 
et puncto B in E puncto posito recta autem BF in EZ 
etiam F punctum in Z cadet, quia BF = EZ. ad- 
plicata iam BF rectae EZ etiam BA, FA c\xm Ejd, 




Vm. Boetius p. 380, 24. 



Sva£ V. 14. ^xy ds] om. Proclus. 19. tds] om. Pbp. 

dvei V. 21. Bri Ar F, sed A eras. 25. tov fisv] filv 

Tov B. 29. dij] di Bb. in^] in ras. m. 1 P. 




ETOIXEISiN 



iipttp(i60ovei. xtt\ ttt BA, rA ixl raq Ed, ^Z. ti yctfi 
^dcis [tiv j] Br i^l pdeiv rijv EZ iqiaQiioOei, aC Se 
BA,Ar nXsvpal inl Tag E^, ^Z ovx ifpaq^Qeaveiv 
dXktt xaQtt}.?.tt^oveiv Bsg at EH, IIZ, avara&ijeovTai 
5 ixt tijs avTrjg ev&eiag Svo Tatg avTat; ev&Eiaig ak^ai 
dvo Bv9ftai teai Bxari^a ixuti^a Jt^as aXXa xal alXrn 
aijntici inl r« avta iibqt] zd avtd aisfata ixovOai. ov 
ewietttvtai di' ovx Kpa ifpapjio^ofiivt^s t^g BF ^d- 
Oftog tJtl Ti]v EZ ^daiv ovx iq^uQfioeovai xal at BA, 

10 AP jiXBVQal inl r«s Ez/, ^JZ. i<pap(i6eovei.v aqa' 
to0Ti xttl yavia tj vao BAT ini ymviav Ti[v vno 
EJZ iipaQfioesi xal fOij avtij setai. 

'Eav aga Svo tifiyava tdq Svo itXsvQdg [tafs] Svo 
xi.svQais leag ixS ^"«rEpKw fxatsp^ xal tijv ^deiv 

15 tfi ^dest ro^v ^XVj "*" ^'J^ yoivittv r^ ymviu fejjv 
f|« T151' vno tav letav tv^simv itsQtE%o(iivriv' ojrsp 

mi stt^tti. 



1 



T»)v So&tteav ytaviav sv&vypaiifiov 8i%a I 

20 XSfl.SlV. 

"Eazo 7] Sod^stea yavia sv0^vyQtt[ifiog ^ vno BAP. ] 
dst Stj avtijv Siy^a TS[isiv. 

EiXii^&ta inl t^g AB ruj^ov arjfistov %6 A, xal ■ 
a^jjjpjjffdo Kito T^g AF Tfj AA Tojj ^ AE, xal ixs- \ 
26 ^v%&a> 7) ^E, xul evvtaTdtca ial t^g ^E tffiyai 
leoTtXsvQov to ^EZ, xal instsvx9-ii) ij AZ' ksyto, ow j 
^ UJTO BAF yavia Si^a T^rfMjrwt v7to T^g AZ CS- I 
^eiag. 

1. itpagiioaovaiy P. BA. FA] PBbp; BA, AT V O i 

corr, ; utenm praebeat F, diaceroi nequit, 8, avvlatuxai p. ] 

9, ^qpaejidooticTii' PF. at] aupra m. rec, P. 10. igiac J 



ELEMENTORUM LIBER I. 29 

^Z congruent. nam si basis BF cam basi EZ con- 

gruet, latera autem BA,Ar cum EAj AZ non con- 

gruenty uerum extra cadent; ut EHj HZ, in eadem 

recta iisdem duabus rectis aliae duae rectae aequales 

altera alteri constituentur ad aliud atque aliud punc- 

tum ad eandem partem eosdem terminos babentes. 

sed non constituuntur [prop. VII]. itaque fieri non 

potest; ut basi ^F ad basim EZ adplicata non con- 

gruant etiam lateraB^^^F cum EA^JZ. congruent 

igitur. quare etiam angulus BAF cum angulo EAZ 

congruet et ei aequalis erit [x. ivv, 7]. 

Ergo si duo triangult duo latera duobus lateribus 

aequalia babent alterum alteri et basim basi aequa- 

lem babent; etiam angulos aequalibus rectis com- 

prehensos aequales habebunt; quod erat demonstran- 

dum. 

IX. 

Datum angulum rectilineum in duas partes aequa- 
les diuidere. 

Sit datus angulus rectilineus BAF. oportet igitur 
eum in duas partes aequales diuidere. 

sumatur in AB quoduis punctum ^, et ab ^JT 
rectae AA aequalis abscindatur AE [prop. III], et du- 
catur jdE, et in jdE construatur triangulus aequilate- 
rus ^EZ [prop. I], et ducatur AZ. dico, angulum 
J3^P recta ^Z in duas partes aequales diuisum esse. 



IX. Simplicias in phys. fol. 14. Boetius p. 381, 1?. 

(loaovciY. 11. in^] supra P. 13. taig] om. Pp. 14. 

T^ P«aBi triv pdaiv P; corr. m. 1. 19. sv^vyQa(U(iov ymviav 
Proclus. 23. in£] ya^ ini P; dici V, corr. m. 1. 27. yeo- 
via] om. BF. 



30 STOIXEIJlN a'. 

'Extl yitQ t-ijT) imlv i) Ad r;} AE, xotv^ &\ ^ 

AZ,, Svo 6ii al ^A^AZ. HvaX ittfs EA^ A2. Ceai si' 

alv maxiiftt ixardQa. xal ^ixei^ ^ ^Z ^aati ttj EZ 

iffij iazCv yavia apa fj vxo AAZ ytovla Ttj vko EAZ 

6 roij iariv. 

'H ap« So9tt6a yavitt tv&vy^ajifioq rj vnh BAV 
dixK TfiftijTfft vno T^g AZ ev9iiag' oxtQ iSn. 



10 Tijv 6o9fiaav tv&tiav 7tC7Ci^a6(iivrji' t 

TEHBiV. 

"Eaxa fj do&stsa ev&eta xsjttQaoiiivi} rj AB' Sit 
Sii rijv AB ev^^elav x£vt£gaa(iivt]v Si%a tbhsIv. 

21vvBeT«Tca i^ avT^s tQLymvov Ca6ni.EV(fov t^ 

iri ABr, xal TBTji^aQco ij vjtb AFB yovia Si%a t^ F^ 

fv&titt' Kiyat, oxi 17 AB tv&sia Si^a riTfirjzai ; 

ro A arifitiov. 

'Emtl yttQ fffij iarlv r) AF rt^ FB, xoivij Sh ^ T^, 

dvo Si] aC AF, P^ Svo rai^ BF., F^ Hem tCalv. 

20 ixttTtQa ixariff^' xal yavia i) vtiq AV^ yavCtf tij; 

vab BT^ 'iafj ietiv ^daig a^a ij AA p&Sti tfj B^ 

iffi) iativ. 

'H apa So9ttaa tv9tta atTtepaafiivti ij AB 61%« 
ritftt^ai xara ro ^' ojicq ISti notiiaai.. 

i. {cziv} PF (in b V ex&a.}; iari ntilgoi comp. B. 12. ij] 
om. bpi m. 2 V. 13. tv9tiav jttneQaaiiivTiv] P; oro. Theon 
(BFVbpl 16. ArB] ante F roe. 1_ Utt. F; TB in raa. V. 
Aute et poat r^ ras. P, sicnt poat tv^Eia \m. 16. 17. t«!T 

i6v comp. V. 19, SvaCv V; fliio 1«« BF, VJ om. b (if 

713 iS m. 2). 21. eW»] Url Vp; comp. Bb. B J] * 

rfta. m. I P. S4. T(f(f ijTOK p. novi\aui\ SU^ta P, mg. n: 
•tq. nmiicai. 



ELEMENTORUM. LIBER I. 



31 




nam cum AA = AE^ et AZ communis sit, duae 

^ rectae jdA^ AZ duabus EA^ AZ aequales 

sunt altera alteri; et basis jdZ basi EZ 

aequalis est. itaque L AAZ^ EAZ 

[prop. vin]. 

Ergo datus angulus rectilineus BAF 
recta AZ in duas partes aequales diuisus 
est; quod oportebat fieri. 

X. 

Datam rectam terminatam in duas partes aequales 
dittidere. 

Sit data recta terminata AB. oportet igitur rectam 
terminatam AB in duas partes aequales diuidere. 

construatur in ea triangulus ae- 
quilaterus ABF [prop.I], et angulus 
AFB recta Fjd in duas partes ae- 
quales diuidatur [prop. IX]. dico, 

rectam AB in puncto jd in duas 

^ partes aequales diuisam esse. 

nam cum AF = FBy et FA communis sit, duae 

rectae AFy JTz/ duabus J5F, FA aequales sunt altera 

alteri; etL^FJ = Br^d. quare AA =BA [prop. IV]. 

Ergo data recta terminata ABiu puncto z/ in duas 

partes aequales diuisa est; quod oportebat fieri. 




X. Sext. Emp. p. 719, 26. Simplicius in phys. fol. 114^ 
Proclus p. 204, 19. Boetius p. 381, 2? 



STOIXEinN a'. 



Tfi So&BtO\i ev&siu dxb Toi- Jtpog «vtjj do- 
^ivtos St](iaiov wpos og&ag ycavlaq ev&tiav 
/pttftft^v ayayetv. 
3 "Eeta r] fihv So&eiSa ev&eia ^ AB to di tfodiv 
tfijftfEow ea avrijs tb F' Set d^ anb tov F ffiifiEiov ■ 
tfi AB Bv&eCft jrpog op^iis yavias evQ-etav yQtt(iniiv\ 
ayayetv. 

EiX'^tp9e> tJtt r^g AF Tv%bv Ot}ft£!ov tb ^, xal 

Hi xe{a9a TJ7 FJ l'er] rj VE, mi ffvvEffraro inl r^g jJE 

tQiyaivov iooitXevQov ro X^E, xal ine^evx^-ai ij ZF' 

}.iyco, ori ty do&eiey ev&eia tj; AB anb tov n;pos 

«vr^ So&ivtos ffijfiEtou rot' F ffpog dpfrag ycovias ev- 

Q-eta ypa{ifi,ij jjxrat ■^ ZF. 

b 'Enel yap fffjj ietlv •^ ^F tij fjE, xotv^ ds ij fZ, 

tftJo ^57 af ^r, rz tfuel rars ET, FZ ieai eialv exa- 

tiqa exaziQO.' xa\ ^deis '*l ^Z ^K6tt trj ZE /■ffTj iativ 

yavia flp« r) vao /JFZ ymvia r^ vjtb EFZ Catj ierCv' 

xai eiaiv irpe%iis. otav 61 ev&eta iz' ev&Eiav ata- 

&Biaa tag iipe^ijg yoviag Haag uA/ijAacg iroi^ 

ixatipa rmv teiav yavi^v iatiV 6(f9i} ap« iarlv it 

riQtt tSv VJto -4rz, ZFE. 

■ Tfi a^a do&eiaij ev&eia rfj AB ditb row «pog «vryj 

I 6o&ivTog eriiLeiov tov r zpbg 6@&ks yaviag £v%eta 

H 25 ypKftft^ ijxtai f} rZ' onsQ eSit aotijeai. 



10. rj] J bx raa. eat in bi JT in ras. V. 18, art^'!- 

F et B ra. 1 (corr. m. 2). aodeftos] -f»- in raa. eflt in V. 
14. yeo(i("i] ei yeaftfiiji V. Zr] rz p efc P oorr, ex ZT. 

16. ^kf/^ rZ] mg. m. 2 P. .lJT] in raa. P. 16. JT, 
rZ\ J ei Z eraa. F; ZT, FJ B, 17. ^aii'»'] P; ^«1/ uulgo. 
nt lin. 18, 19. iins V; corr. m, 2. 23. tnl (alt) ij V; 

corr. m. S. ^B] in raa. P. 



ELEMENTOBUM UBER L 



33 



XL 

Ad datam rectam a dato puncto in ea sito rectam 
perpendicularem erigere. 

Sit data recta AB, punctum autem datum in ea 
situm r. oportet igitur a F puncto rectae AB per- 
pendicularem rectam erigere. 

sumatur in AF quoduis punctum A, et ponatur 

FE = rj [prop.II], et in AE 
triangulus aequilaterus constru- 
atur ZAE [prop. I], et duca- 
tur Zr. dico^ ad datam rectam 
jff_AB 'fk dato puncto in ea sito 
^ r perpendicularem erectam esse 
rectam lineam ZF. 

nam quoniam jdF = FE et communis FZ, duae 
rectae jdF, FZ duabus EF, FZ aequales sunt altera 
alteri; et basis AZ basi ZE aequalis esi itaque 
LAFZ = EFZ [prop.VIII]; et deinceps sunt positi. 
ubi autem recta super rectam lineam erecta angulos 
deinceps positos inter se aequales efficit, rectus est 
uterque angulus aequalis [def. 10]. itaque JFZ^ ZFE 
recti sunt. 

Ergo ad datam rectam AB a, dato puncto in ea 
sito r perpendicularis recta linea ducta est FZ] quod 
oportebat fieri. 




XI. Boetius p. 381, 4. 



Enclides, edd. Heiberg et Menge. 



'£jri Trjv So%^sl(Sav EV&Etav aicsLQOv axo xov% 
So&dvtos GrmtCov, o ff^ iaziv in' auT^g, xrf-: 
^erov £V&Eiav yQafi.[i7}v ayayetv. 

"EffTOJ ri [ih' Sod-siaa tv&sia aaeiQOs ^ j4B to S\ | 
dofrsv eri[i£tov, ftj] iariv iit' avt^s, to F' Set Si^ j 
iiii TJjv So&Eieav sv&etav a^BiQov xriv AB «a;o tou ] 
Sod^dvTos 0f](isiov rov F, o (t^j ^ffiti/ ^jc' avT-^g, xd&etov I 
*uS'erai' ygafifiijv ayccystv. 

EtXriip%co yag inl za tTetfa ^tpJj rijs AB fd&sCa^ I 
Tvxbv ari^iZQV xo ^, xal zsVrpp ^lv Ta F (Jtatfrij- 
(tKTt di Tpi r"^ Kvxlos yEy^dfp&m b EZH, xal te- 
TfHjffd^ti) ^ £if £U'd'*ta d('j;a ftaro; ru ©, xai iat^8vx9to- 
tJav ttC rH,r&,rE Ev&Etat- i.dy(o, oTt inl rrjv So&et- 
15 aav Ev&eittv aneiQov tjjv AB anb xov So&ivrog • 
arjfieCov Tov F, o ftij iariv fV «wT^^g, KaS^ETOs Tjxrat 1 

17 r®. 

'EitEi ywti terj iatlv -ij H& ti] ®E, xoivr] Se ^ 1 
&r, Svo dii aC H&, @r Svo rafg E®, ®r taat elalv 

80 ixatifftt exttreQtt' xal ^aaig i] FH ^doei ttj TE iariv 
Car]' yavCtt aga 7j vao r@H ymvCa Tf; vao E@r iativ 
taij. xaC eCaiv ^qsel^g. orav Se Ev&eta iit' ev&etc 
ata^eiaa tas i(pE^^s yavCas iOas ttkXr]kttis Jtot^, op^ j 
ixateQa t(dv ftfrav ytovimv iaxiv, Koi 55 i^earrjKvta bv~ 1 

S5 ^tta xd&ETOs xaXEtrai i<p' ^v icpeafrixEV. 

'Ead r^v So9etattv «pa evQetav aneigov tijv AB \ 
dno roC So^ivros arjfieCov Tow F, o (iij iotiv in' 1 
Tf]s, xtt&itog ^xiai ^ F®' oreip i^ffft itoiijaai. 

2. Ante am! tas. 2 litt. P. 9. yjajip.fjv] mg. m. recenti 
V. ^ll. fiiv] aupra m. 1 P. %ivte<o tra F moI JioDr^;iB« 

BPbp. 13. fv^iitt] P; om. Theon (BFVbp). U. TE] e 



A^ 



ELEMENTORUM LIBER I. 35 

xn. 

Ad datam rectam infinitam a dato puncto extra 
eam sito perpendicularem rectam lineam ducere. 

Sit data recta infinita j^B punctum autem datum 
extra eam situm F. oportet igitur ad datam rectam 
infinitam jiB b, dato puncto extra eam sito F per- 
pendicularem rectam ducere. 

sumatur enim in altera parte rectae AB quoduis 
punctum ^, et centro Fradio autemF^^circulus describa- 

turiSJZif [afr.S], etrecta EH 
induas partes aequalessecetur 
[prop. X] in B, et ducantur 
rectaerif,r©,rfi:. dico,adda- 
tam rectam infinitam AB a dato 
puncto r extra eam sito per- 
'^ pendicularem ductam esse FS. 

nam cum HS = @Ej et communis sit Sr, duac 
rectae H0, &r duabus ES, &r aequales sunt altera 
alteri. et basis FH basi FE aequalis est. itaque 
L r&H = E@r [prop. VIII]. et deinceps positi sunt. 
ubi autem recta super rectam lineam erecta angulos de- 
inceps positos inter se aequales efficit^ rectus est uterque 
angulus aequaliSy et recta linea erecta perpendicularis 
adpellatur ad eam^ super quam erecta est [def. 10]. 

Ergo ad datam rectam infinitam AB bl dato puncto 
r extra eam sito perpendicularis ducta est FS] quod 
oportebat fieri 

Xn. Schol. in Archim. III p. 383. Boetius p. 381, 7. 

corr. m. 2 P, E dub. in F. sv^stdi] P; om. Theon (BFV 

bp). 16. ndftsxos] ante x ras. V, ut lin. 28. 19. 0r] TS 
BP. H©, 9r]er, BH e corr. P; re, GH B; H et r 

eras. P. dvei BF. 

3* 




ETOIXEIilN c 



noi^, ^TOt Avo OQ&ag ij Svalv OQ&ats Ceag «o 



5 Ev&£ta yag rig fj AB ia' sv&-Etav rijv Fid eza- 
&£i0tt yavias noiBirai ras vao FBA, AB/d- kiyo, on 
ai VXQ FBA, AB,d yaviat ijtoi dvo OQ&ai eittiv ^ 
3v0iv opd^atg taai. 

Ei (ihv ovv l'6Ti iatlv ^ vno VBA r-^ v%h ABJ, 

I) Svo QQ^ai Eieiv. ei 61 ov, ^x^*^ ^^° '^^^ -^ ari^eiov 
Trj r^ [Bv&^sia] jrpos o^d-ag ri BE' «t apa vno PBE, 
EBA dvo dp&ai dat.V kuI iael tj vwo VBE SvbI rais 
imo FBA, ABE ieri ioriv, xoivri 7CQoexeia&at ij vxh 
EBJ- al aga vno FBE, EBd XQieX rafg vno PBA, 

5 ABE, EBA tsai tieiv. itaKLv, inel rj iuto ABA Svel 
■cccis vno ABE, EBA layi iexiv, xoivii jrpouxEtfffl-ra ij 
vx6 ABV' al ap« vno JBA, ABF iQiel ratg vao 
^BE, EBA, ABr teai eleiv. iSsix^euv Sl xal a( 
vni) FBE, EB^ TQtal zats avtatg taai' za S\ Tp 

avrm iaa xal «AAjjAotg *'ffrh' faa' xal aC vxb FBE, 
EB^i^ «pa xats vxo ABA, ABF laai. eleiv alla 
al v7to FBEjEB^ Svo OQ&ai tlOiv Koi aC vno^ABA, 
ABF apa Svelv OQ&atg taat aleiv. 

'Eav aga tvd^eta iit' tv&etav Graifetaa ytoviag xotfj. 



I 



a. 'Eav] P m. a, Proclna p. 292, 16, PMlop. in anal. U; 
in Y E rubro colore postea additum, ut eaepe in hoc oodioa i 
Utteiae tsiti&les, a in raa. (sed lin. 84 ag av); oxav P m. 1, 
Pbilop. iu phjH.; mb Sv Theon (BFbp, PaellnB et aine dnbio 
V m, 1), ProoluB errore libratii p, 291,20. 8. avtriv} Ho ' 

Piocliu. 10. ov] post raa. 1 titi Y, 11. Ev»ii<f] P mff. < 

m. 1; om. BFYbp. 12. tlotv] P, ctai uuko. 18. io^iv] 

P, ^or^ udlgo. 14. zgiai] ex. zi)Uiiv m. 8 P. 16. «^oAj 




ELEHENTOBUM LIBER I. 37 

• XIII. 

Si recta super rectam lineam erecta angulos effe- 
cerit, aut duos rectos aut duobus rectis aequales an- 
gulos efficiet. 

nam recta aliqua AB super rectam Fz/ erecta an- 
gulos ef&eini FBj^j ABjd. dico, angulos FBv^, ABjd 
aut duos rectos esse aut duobus rectis aequales. 

iam si FBA = ABjd, duo recti 
sunt [def. 10]. sin minus^ a B 
puncto ad rectam Fjd perpendicularis 
ducatur BE [prop. XI]. itaque FBEj 
EBd duo recti sunt. et quoniam 
FBE = FBA + ABE^ communis 
adiiciatur EBjd. itaque FBE + EBjd -^ FBA + 
ABE -}- EBjd [x. ivv, 2]. rursus quoniam jdBA = 
jdBE -|- EBAj communis adiiciatur ABF. itaque 
jdBA 4- ABr= ABE + EBA + ABF [id.]. sed 
demonstratum est, etiam FBE-^ EBA iisdem tribus 
aequales esse. quae autem eidem aequalia sunt, etiam 
inter se aequalia sunt [x. ivv. 1]. quare etiam 
FBE + EBA = ABA + ABT. 
uerum FBE + EBA duo recti sunt. itaque etiam 
jJBA -{- ABF duobus rectis sunt aequales. 

Ergo si recta super rectam lineam erecta angulos 

XTTT. Simplic. in phys. fol. 14. Philopon. in phys. h IIII, 
in anaL n p. 65. PBellus p. 36,40. Boetius p. 381,9. 

sl4fi PBV; comp. b. 16. tinj] corr. ex Ua V. ictip] PP, 
comp. b, icz£ unlgo. 17. &Q€ij a^a ycDviai (in ras.^ at V. 

20. xa/] (alt.) post ea add. V; in mg. add. m. 2: ai dvo. 

21. siaiv taai p. 22. siaiv] PF; comp. 6b; siai, uulgo. aC] 
om. V. 23. tt^a] om. BF. 24. 'Edv] «g at^ PBPVbp. 



38 ETOIXEiaN u. 

rjtoi Svo ofiQ^as ij Sv6lv oq&atg taaq Jcoti^aBi' oJrfip 
idu Stt^iai,. 

tt)'. 

'Eav «Qog tivi ev^bC^ xal rm rapog «wr^ ffij- 

6 fiEiip Svo ev&etai fi^ ixl ta avza (ii(f7i xe£- 

[isvttt .Ttts i^E^ijs yatvtttg Svolv o^&afg tOttg 

TtotaOiv, in' EvQ^Eiag ieovtai aXliilaig aC ev' 

&Ei:at. 

ilpos yd(f tivi Ev&Bt^ ry AB xal ta jrpog avr^ 

JO iSr^ftEl^ T^ B Svo EV&ettti atBr,B^iJ.ii iitl ta avxtt 

nif/ij XEifiEVRt rag ig)£^^s ycJviag tug vxo ABF, ABd 

Svo OQ&alg taag noisitcoSav' Xiya, oti iic ev&eius 

ictl Tfj rS T} BJ. 

El yap ftij ieti. tij BF ix^ Evf^Etas ti B^, isrto 
IB tj} FB iit' EV^Biag fj BE. 

'Eael ovv ev&eta 17 AB ia Ev&etav c^v PBE 
itpiarijxBV, at «pa vxit ABF, ABE yaviat dvo oq- 
9atg tOttt eiaiv Eial de xal at vno ABF, ABJ 8vo 
off^alg taaf at aQa vito VBA, ABE tatg vnh FBA, 
AB^ teat elsiv. xoivij dqj^p^^a&ai ij v«6 FBA' Aoin^ 
aQtt ij va6 ABE lomii tij vjio ABA i6ztv tOri, rf 
iXtteaatv zy (leiiovf ojt£p iozlv advvazov. ovx apa 
i%' Ev&eiag iazlv 17 BE tij FB. oykoimg S^ SEi^OfiEV, 
oti ovSh ttlKti ris alijv f^g B/d' iit' ev&eias a^a iatlv 
8B ii PB tij BA. 



1. ojtee tdei Sti^ai] :— BFVi oin, bp; Seiiai mg. m. 2 
FV. 2. atilat] «oi^aai P, corr. m. 2. 4. fv^titf yqaiLy.^ 
F. 5, Ev&Ffat e^^e ProcIuB; cfr. p. 296, 17. viCfie*ai\ om. 
ProcluH. 6. Svaiv] Svo ProcluB. 13. laziv F, nt Lm. 14.' 
U. Brj coir. BS rs V. 16. VB] BTb. 17. ui] i\ 

cow. B. avaiv V. 18. tlaXv Si P. Svaii/ V. IB, (op- 
%uti — 20. tlalv] poatea add. in V in imo folio. 20. slalv 



l 




ELEMENTORUM LIBER I. 39 

e£fecerity aut duos rectos aut duobus rectis aequales 
angulos efficiet; quod erat demonstrandum. 

XIV. 

Si duae rectae ad rectam aliquam et punctum eius 
non in eadem parte positae angulos deinceps positos 
duobus rectis aequales effecerint, in eadem erunt linea 
recta. 

Nam ad rectam aliquam ^JB et punctum eius B 
^y y duae rectae JBF, JB^ non in ea- 

dem parte positae angulos deinceps 
positos ABryABjdl duobus rectis 
T B ^ aequales efficiant dico, FB et 

B^ in eadem recta esse. 

nam si JBF et B^ non sunt in eadem recta, FJB 
et ££ in eadem recta sint. 

iam quoniam recta AB super rectam FBE erecta 
est, LABF-^-ABE duobus rectis aequales sunt 
[prop.Xni]. uerum etiam ABr-^ ABA duobus rec- 
tis aequales sunt. itaque FBA -^- ABE — FBA -|- 
ABA [x. ivv, 1]. subtrahatur, qui communis est, 
L FBA. itaque L ABE = ABA [x. ivv. 3], minor 
maiori; quod fieri non potest. quare BE ei FB non 
sunt in eadem recta. similiter idem de quauis alia 
recta praeter BA demonstrabimus. itaque FJBetJB^ 
in eadem recta sunt. 



XIV. Simplic. ad Arist. de coel. fol. 131\ Philop. ad anal. 
n fol. 4^ Boetius p. 381, 11. 

PF; bIoI Qnlgo. %oi.vq — 21. xi vno\ in ras. in Bununa pag. 
V. 21. Xoini\ Xoi V. 22. AaiTTov F. 23. FB] BF P, 
et V sed corr. 24. ovd* p. 26. xjj] sequitor ras. 1 litt. 
in V, x^g comp. b. 



4i0 ETOrxEinN «'. 

'Eav aga rapdg Tivt sv&fia xal rp jrpog «t^TTj ff^- 
/iE^cj tfuo EV&eTai ftij ^wl ra «utk ^Epl XtifiEvai tag 

^^iel^S jimvtag Svslv OQ&ats fo«S KOttoOtv, iw' Evdfiag 
Seovrai «AA)]A«is «f sv&Etai,' ojteg ^dev Sai^ai 



'Eav $vo Bv&etai tifivaeiv aX^^Xaq, zag 

xuTa X0Qi'q)riv ytovCaq ^aaq aXX^kats xoiovtSiV. 

^vo ywp EV&iZai at AB,r^ ttiivhaeav aXliq- 

Xag xata tb E srjiiitov X^ya, oTi iatj iatlv 17 [liv 

10 vno JEV yavia t^ vno jdEB, ^ Si vao FEB ty 

vno AE^. 

'Ensl yitQ ev&Bta ij AE i% Bv%hlav xr(v T/i i 
BStt\XB yaviag mtovoa tag vxo TEA, AE^, aC aqa ' 
V7C0 FEA, AEA yaviai dvolv OQ&atg t^dat Bioiv. 
16 Xiv, iitel BV&Eia 7) AE i%' BV%Eiav Tijv AB ifpdOtTjXB 
yaviag aoiovOa zag vitb AEA, AEB, at apu iino 
AEA, AEB yaviat Svolv O0ft«ts toai Bioiv. iSsix^-ij- 
aav Si xal at vab PEA, AEJ Svslv off&aig tOaf 
ttt UQa vao FEA, AEA xats v^itb AEA, AEB lOai. 
20 Biaiv. xoivf) dipfip^od'a} ri V3t6 AEA' ^otni; aQa 7} vxb 
VEA kotxy T^ vTibBEJ fffij iatCv oy^oitoq Sii Sbix- 
•&)jff6T«i, oit xal at vitb FEB, AEA tSai Bleiv. 

^Ekv a^a Svo Bv9Bi:ai. tiftvmOiv aXl^Xag, tag xaric 
xo<fv<pi}v yaviag teag alX^^Xatg aotoveiv oxbq idet ' 
^35 SBt^ai. 



4. at] om, V, 7. aoiovaiv] TioiiMii» Pro 
{nel -a.) eodd.i ofr. lin, 24. 13. Iqiiei^nMv BI 
18. Se9ais] in rat. V. 14. elaiv]PBF; comj 
15. in'] hi Pb, lq,iatn%tv PBP. 16. «f 
"TBl mg. m. 1 p, 19. aga] om. F. -catc. 
eiaiv] FF; comp. b; ilal aiilgo. 



16. al aoa ixo AEJ, 

■catg] aea xais " 
aqir)f>lia&ie V. 



^te 



ELEMENTORUM LIBER I. 41 

Ergo si duae rectae ad rectam aliquam et puuc- 
tum eius non in eadem parte positae angulos dein- 
ceps positos duobus rectis aequales effecerint, in ea- 
dem erunt linea recta; quod erat demonstrandum. 

XV. 

Si duae rectae inter se secant^ angulos ad uerticem 
positos inter se aequales efficiunt 

Nam duae rectae j4Bj FJ inter se secent in puncto 
E. dico, esse L ^EF = JEB et L ^EB = AEJ. 
nam quoniam recta AE super rectam FA erecta 

est angulos efficiens FEA^ AE^, 
anguli FEAy AEA duobus rectis 
aequales sunt [prop. XUI]. rursus 
? quoniam recta AE super rectam 
AB erecta est angulos efficiens 
B^ AEA, AEB, anguli AEA, 
AEB duobus rectis aequales sunt [id.] sed demon- 
stratum est, etiam angulos FEA, AEA duobus rectis 
aequales esse. quare FEA + AEA = AEA -^- AEB 
[x. ivv. 1]. subtrahatur, qui communis est, L^EA. itaque 
FEA = BEA [x. iw. 3]. similiter demonstrabimus, 
esse etiam L TEB = AEA. 

Ergo si duae rectae inter se secant, angulos ad 
uerticem positos inter se aequales efficiunt; quod erat 
demonstrandum. 




XV. Boetios p. 381,15. 



rJE^l litt. EA in ras. V. BEd^ JEB B et in ras. V. 
^ij] ai b, et V m. 1 sed corr. 24. noimaiv F, 



43 



STOrXEIfiN D 



2t 



[IIoQtSfia. 

'Ex dij tovTov ipKVEQov ort, iav Svo ev&Btai zi- 
(ivmSiv dXk^las, ras ^gis t^ '^o(*H ycoviag xix^aoiv 
OQ&atg taag aof^ffovaiv.] 

6 ts'. 

Tlavxoq XQiycovov iiiag riav nlBVQav itQoa- 
fx^Ajjfl-stffTjs fi ixTos yavia ixatdQag xmv ivtog 
xal ansvavxCov yavttav fieC^av isxCv. 

"Eaxia XQCyiavov xo ABF, xal aQO0exps§?ii]a9a} av- 

roj; fita nlsvQa ij BF ial to z/" Xiyta, ort 17 ^xros 
yavCa ^ vno AF^ fidC^av iexlv ExatiQag tmv ivxos 
xal ansvavtCov tiav vno FBA, BAF ytaviav. 

TiTftijfffrra ^ ArSC^a xata to E, xal iai^evx^-Etect 
fj BE ix^E^X-^e&oj in' sv&BCag ial ro Z, xal xtCaQ^a 

5 Tfj BE terj rj EZ, xal iTts^evx&ia n ZF, xal Strn^m 
71 Ar Inl xo H. 

'Eatl ovv Tffjj iatlv ^ fiiv AE t^ EF, 17 Sl BE 
rg EZ, Svo 3>) at AE, EB Svel xalg FE^ EZ teai 
eialv ExaTEQa ixatEQa' xal yavCa tj vao AEB yavCif 

) TjJ v%h ZEV i'eri ietCv xata Kopr^J^i' yaQ' ^aatg 
KpK f} AB ^aast ttj ZT tai\ iatCv, xai x6 ABE xqC- 
yavov xa ZEF TQiymva iezlv feovj xal at lomal 
yavCtti xatg loinatg ymvCaig leat eCalv exazeQa ixa- 
XEQa, vip' ag at taai TtlevQal vnoteCvovaiv ' Cer] ana 

5 ieziv ri vnb BAE tft vno EFZ. (teC^iDv Si iativ ij 



PVb et alter eodex Grynaeij 

1 B in imo mg. m. 1; habent F, Pro- 

lag. m. 2 legitur cum altero eod. Qry- 

tfavsQov, oxi iav iaaiS rtJiozovv ev&etat xi- 

i, _; — .; f — leaaagiiiv oe^ait 



1. ■aoi/iaiia 
ta p legitur j 
oln», PBellQS p 

niiei: i% SijTC , .^ , 

[ivoietv all^lae, ras "Qot tri lofi^ ^eovCo 



f(WB iroiiiiiofai; idem mg. m. 1 praebent F {tizgaeiv, bdiiJ- 
aovetv) et b {tittaeaiv, nofqaovatv) et habuit PselloSi Procloa 




ELEMENTORUM LEBER I. 43 

XVI. 

In quouis triaDgalo uno latere producto angulus 

extrinsecus positus utrouis augulo interiore et oppo- 

sito maior est. 

Sit triangulus ABFj et producatur 

unum latus eius jBF ad ^ punctum. 

dico esse LAr^>rBA et 

Ar^>BAr. 

secetur AF in duas partes aequales 

in J?[prop. X]; et ducta JBEproducatur 

in directum ad Z, et ponatur EZ =BEf 

et ducatur ZF, et educatur AF B.d H. 

iam quoniam AE = EF et BE = EZy duae rec- 

tae AEj EB duabus FEy EZ aequales sunt altera 

alterL et L AEB = ZEF (nam ad uerticem eius est) 

[prop. XV]. itaque basis AB basi ZF aequalis est et 

£s. ABE^ ZEFy et reliqui anguli reliquis aequales 

Bunt alter alteri, sub quibus aequalia latera subten- 

dunt [prop. IV]. itaque LBAE^ EFZ. uerum 

^XVL Schol. in Pappum III p. 1183,4. Boetius p. 381, 17. 

p. 305,4 de 8U0 adiicit. praeterea in V mg. m. 1 reperitur: 
noQ^licc, i% dri tovtov qfuvsQov^ oti iav oauiSi^noxovv svd^BCcci 
xinvoDaiv ttlXriXag rag itata %0Qvq>7iv ymvias Caag dXX^qXatg noi- 
iqaovaiv. ZambertuB nullum omnino porisma habet, Gampanus 
id, quod recepimus. 2. tifKoaiv p. 3. nQog tij tofi^] Bp; 
xittaQag Proclus. at nQog tij tofi^ ymvCai F. titQaaiv] 

BFp; titxaQaiv Proclus. 4. iaag\ taai F. noiriaovaiv] Bp; 
noiovaiv Proclus; slaiv F. 6. tmv nXBVQmv] nXBVQag Proclus; 
tmv nXBVQug V, sed corr. nQoa- e corr. V. 7. tov tQi- 

ycivov ymvCa Proclus. 8. dnBvavtCmv B. ycovKov] P, Boe- 
tius, Campanus; om. Proclus et Theon (BFbp; in V comp. 
add. m. 2). 12. dnBvavtCoiv B. 14. Post BE ras. 2 litt. 
P. in Bv^BCag] P; om. Theon (BFVbp). 16. H] K in 

ras. p. 20. iatCv] comp. b; iatC BF. 21. iatCv] PF; 

comp. b; ictC uidgo. 25. fis/^os P, corr. m. 2. 



44 rroiXEiiis n'. 

vjtoErd tiis 't>^o BrZ- (lei^iav aga ^ -oreo jflT 
T^g vjia BAE. 'Ojioicag Si} rijg BF rft^jjfi^vi/g dix"^ I 
dtix&fjeittei xal ij v:t6 BFH, Tovregtiv tj vjto Ar^, • 
jiiC^mv xal tijg V3i6 ABF. 
I TlKvrog «pa ipij-ravou (ii.ag rtav nkEVQav HQoOm- I 
fiXri&sietjs Jj ^xrdg yavCa ixati^ag rav ivrog xal aa- 1 
EvavrCov yatviav fisCimv iStCv onep £3ei Seti,ai. 

Ilavrog rQiydvov at 8vo ytovCai dvo op- 
I #£01' iXasoavis eCst aavrrj nerai.aii^av6fievai, 
"Eoria tQCyavov to ABF' kiya, ori roij ABrxQi- ] 
yoivov at Svo ytavCai Svo op&iov iXdrroiiig tlGi Jldvzjj 1 
HiraXaii^ttvofifVRi. 

'Ex^B^^e&a yuQ ^ BT inl rb J. 
\ Kal insl TQiyavov tou ABF ^xrog iori yavCa ^ 
vab AF^^., fisitfnv ierl r^g ivrog xal aitEvavrCav rijs 
vxb ABF. xotvii XQoSxeCs&a ij VTib AFB' al aQa 
vnb Ard^ATB rmv vnb ABr^BFA (leCtov^g eCfftv. 
aXSL' at VTtb ATJ , AFB dvo o^&als teai sCsCv at 
20 «pK Tjjro AB r, B PA 6vo oq&cov ikaseovig cCstv. 
oftoCcas Sii det^OfiEV, ori xal at vitb BAF, AFB dvo 
QQ&wv ildsoovig aCsi xal izt at vab FAB, ABF. 

Tlavrog uQa tQtyfovov at Svo yotviat dvo opftrav 
iXdaoovig eCot ndvtTj iietaXafi^avofiivafonEQ eSei dti^ai, 

1. AFJ] AFJ xai F. 2. S^] BFbp; tfi P et V inser- 

tDin m. 2. TtzfijJiierTii] zjiTfiiiotis B. 6. dnfvavxltov B. 
7. yiovi&v^ P; om. Theon (BFVbp). itlitii] PBp et e corr. 
V; :-^ F; Ttoi^ffBt V m. 1, \i. 10. ileiv P. ^ertcXaii^- 

v6^evai\ -at eraa. V. 13. iXdaaaviq BVb. tCaiv PP. 

15. AEF] BF euan. F. 16, larlv P. antvavTieiv B, aed 

corr. m. 1. 19. SveCv B. eloiv iWi B. 20. HttXTovtt 

F, 21. cno] om. Pp; m. 2 PF. 23. tCaiv PF, comp. I 



J 



ELEMENTORUM LIBER I. 45 

L EFJ > ErZ [x. ivv. 8]. quare L AFJ > BAE. 
Bimiliter recta BF m duas partes aequales secta de- 
monstrabitur etiam L BFH^ ABF^ h. e. 

L Ar^ > ABr. 

Ergo in quouis trianguro uno latere producto an- 
gulus extrinsecus positus utrouis angulo interiore et 
opposito maior est; quod erat demonstrandum. 

XVII. 

Guiusuis trianguli duo anguli duobus rectis minores 

sunt quoquo modo coniuncti. 

Sit triangulus ABF. dicO; 

angulos duos trianguli ABF 

duobus rectis minores esse quo- 

jff J' J quo modo coniunctos. 

producatur enim jBF ad jd, et quoniam in trian- 

gulo ^jBFextrinsecus positus est angulus Ar^diy ma- 

ior est angulo interiore et opposito ABF [prop. XVI]. 

communis adiiciatur AFB. itaque 

AFJ -f ATB > ABr+ BFA [x. ivv. 4]. 
uerum AFA -(- AFB duobus rectis aequales sunt 
[prop. XTTT]. itaque ABF -^- BFA duobus rectis mi- 
nores sunt. similiter demonstrabimus, etiam BAF-^- 
AFB et praeterea FAB + ABF duobus rectis mi- 
nores esse. 

Ergo cuiusuis trianguli duo anguli duobus rectis 
minores sunt quoquo modo coniuncti; quod erat de- 
monstrandum. 




XVII. Proclus p. 184, 1. Boetius p. 381, 19. 

24. kXaxxovsq F. bIoiv PF; comp. b. dct|ai] noiriaai V, 
sed supra scr. dBi^cLi m. 1. 



IlavTog TQiyfavov rj (iti^cav nltv^a rijv fisi' 
£ova yoivCav vxotEivei. 

"Effza yttQ rQiytavov to ABF [iBi^ova ij^ov t^v JtP 
6 iiXivQttv rijs ^B' Xiya, oti xal ytavia 7) vao ABT 
[isi^tjov ierl rrjg vtco BFA. 

'Ensl yicQ {itit,o}v ierlv ^ AF rijs AB, xtic9io TJI 
AB teti ji A^, xal ixs^Evx&in r} BJ. 

Ktd insl tQiyiuvov tov BPjJ ixrog iszi. yavla ^ 
v%o A^B, ftf6£(oi' ietl T^g ivths xal antvavriov t^s 
VTCo iJFB- fffjj dh ij vni A^B t^ vjro AB^, iml 
xal TiXtvQK 5j AB Tfl A^ iariv ieri- ii£it,av aQa xal 
11 vjto ABA tijs VTto AFB' zoXlm «pa ij vao ABF 
liai^av ietl t^s vm AFB. 
5 Ilttvros aga rQiymvov ij (lEi^tav nXsvQa t^v [tti^ovtt 
yatviav vnotsivEV onsg eSei dat^ai. 
t&'. 
navrog tQiycovov vao tijv jiEi^ova yavCav 
71 f»ei£rav TtXEVQtt vitoteivEi. 
"Eera tQiymvov ro ABV [lEitova i%ov t^v v«b 
ABF ytavCnv t^s vaoBFA' liyia, on xtcl xkevQa ^ 
AF TtXEVQug T^g AB (iii^cov ietiv. ' 

Ei yaQ (i)J, ^'roi fffij ^ffTiv ij AP tij AB ^' 
iXttaeetv i^et/ niv ovv ovx ^eriv 17 AF tjj AB' faij 
35 yaQ Kv ijv xal ytavia r] vno A BF r^ vTto AFB' ovtt 
lari Si' ovx aQK fffi? ^flrli' r} AF i^ AB. avSh (t^v 
iltiaecav ietlv ^ AF tijs AB' ilttaatav yaQ av ■^v 

6, iarCv P. 8. koC— BJ] mg. m, 1 P. 9. BPJ] 

PBF; B^r unlgo. 10. AJB] corr. ei ABJ F. ioiA 

P. 11. JrB] Pp; AFB BFb et e corr. V. 12. AB] bd- 
pcft BcriptQin iJ b m. 1. 13, moUto — 14. A TB] mg, m. 1 P. 
14. ^OTi^v P. 16. SsiQ ISci Stiiit'i] om. Bbp; m. S add. V. 



£L£M£NTORUM UBER I. 



47 




xvin. 

In quouis triangulo maius latus sub maiore angulo 
subtendit 

Sit enim triangulus ^jBFhabens -^r> ^5. dico, 
etiam esse L^Br>BrA. 

nam quoniam AF > ABj ponatur AA — AB 

[prop. II], et ducatur B^. 
et quoniam in triangulo BFA 
extrinsecus positus est LAAB, 
erit L^^B>^rB, quiin- 
jp terior est et oppositus [prop. 
XVI]. sed L AjdB ^AB^, quoniam etiam AB = Ajd 
[prop. V]. itaque etiam LABjd>ArB, quare multo 
magis L^Br>ArB [x. ivv. 8]. 

Ergo in quouis triaugulo maius latus sub maiore 
angulo subtendit; quod erat demonstrandum. 

XIX. 

In quouis triangulo sub maiore angulo maius latus 
subtendii 

Sit triangulus ABF habens 
L ABr > BFA. 
[dico, etiam esse Ar> AB. 

nam si minus, aut AF = AB aut 
Ar<AB. iam non est-^J^^^A tum 
enim esset L^Br^AFB [prop. V]; 
jr uerum non esi itaque non est AF = AB. 
neque ueio AFKAB. tum enim esset L^BFKArB 




.? 



XVni. Boetius p. 381, 21. XIX. Boetius p. 381, 23. 



21. BrA] corr. ex FBA b. 
26. ^mv P. 



17] in ras. 3 litt. m. 1 P. 



48 STOIXEIflN «'. 

xal ytavia i] vno ABF zijg vno AFB' ovk lazi Si'\ 
ovK a(/a HaaOfav ietlv 17 AF rijg AB. iStix^ ^^» 1 
OTi ovdi [071 iaxCv. liBi^mv aga iazlv ^ AF tijg AB. 
i7avT0g tt(fa Tgiymvov vn,o Ti\v (ttitova yaviav ij 



IJttVTog zfftydvQv aC flvo zXsvQal rijg Aot- I 

Jt^^g (laiiovig eiet ^cci^r]] ^eraXttyL^av6(i,evai. 

"Etsra )'ap TQiyoivov tb ABF' Xiya, oti roi) ABV ' 

10 TQiymvov ai 3vo nXevpul xijs Aotff^s f^Bi^ovig bCVi 
navTij liETttXafLpavoiievai., ttl (liv BA, AF r^g Bfj 
ttt Si AB, Br Tijg AF, at fii BF, FA rijff AB. 

iJL-^X&m yuQ 7} BA iaX zo ^ eTjfietov, xal xeie&a'M 
Tfj FA foq Ti A^, xttl iae^evji^&a ^ ^F. 

15 'Ejitl ovv tenj ierlv ij ^A r^ AF, teri ierl xal 
yiavia 7} vjtoA^rT^vTCoAr^'y,eii,av aQtt ^ vno BVjd 
t^S vjto A^F' xai inel TQiyavov iari to zJFB fisi- 
^ovtt ^jjov tijv vno BT-d yavlav rijg vsb BdF, vnb 
S% T^v fiei%ova yaviav ij [lei^av nXevpa vitoreivsi, ij 

SO dB aQtt f^g BF ieri ^ti^mv. tenj Sh ^ /JA rfj AF' 
fiii^ovss ttp« «^ BA, AF rijg BF' ofioias Sij Sci^o- 
fiev, oTi xttl at (tlv AB, BF rije PA (teitovig eteiv, 
at Si BF, FA tifg AB. 



XX. Boetiua p. 381,85. 

l. fstiv P. 2. tqe] «S !»■ 3. hz(v\ PFV; oomp. 

b; iatC anlgo. iarlv\ comp, bj laxai F. 4. apa] vui. 

V. 7. xuis lotnafe V; corr. m. 1. 8. tlal\ riatv PF; 

comp, b. 9. 0T(] om. F. loo] e corr. V. 10. ipi- 

yiovov] -ov e corr. V. zait loMafe V, aed corr. tlai\ 

ilaiv PF; comp. b. 11. BT] FB BF, et T oort, ex BP. 
la, AF] jr F. 14. rfi} corr. ex lije V. .JT] TJ F. 



1 



ELEMENTORUM LIBER I. 49 

[prop.XVIll]. uerum non est.^ itaque non est^J^^-^^JB. 
demonstratum autem est^ ne aequalem quidem esse. 
quare Ar> AB. 

Ergo in quouis triangulo sub maiore angulo maius 
latus subtendit', quod erat demonstrandum. 

XX. 

In quouis triangulo duo latera reliquo maiora sunt 
quoquo modo coniuncta. 

Sit enim triangulus ABF. dico, in triangulo ABF 
duo latera reliquo maiora esse quoquo modo coniuncta; 
BA + Ar> JBF, AB+Br>Ar, Br+rA>AB. 
educatur enim BA ad z/.punctum, et ponatur 
^ AJ = FAy et ducatur ^F. iam 
quoniam ^A = AFj erit etiam 
LAJr= AFJ [prop. V]. 
itaque L BFJ > AJF [x. ivv. 8]. 
et quoniam triangulus est JFB ma- 
iorem habens angulum BFJ angulo 
Bjdir, sub maiore autem angulo 
B~ J^ maius latus subtendit, erit ^B> BF 

[prop. XIX]. uerum jdA = Ar. itia.que 

BA + Ar> JBF.O 
similiter demonstrabimus, esse etiam 

AB + Br> FA et Br+ rA> AB. 

1) Nam JB^ JA + AB. 

15. iaz£] comp. b; iativFF. 16. Post AFJ Sidd.ciXX' rj vno 
BFJ ymvitt rrjg vno ATJ fiBLtoav iazC mg. m. 1 V, mg. m. 
recenti p. 17. AJF] corr. ex AFJ F. iativ P. 18. 

BJF] corr. exAJF V; JAB uelJAr F. seq. ras. magna 
P. 20. iativ P. JA] AJF. JA xfi Ar]JB xat^ 

AB, AF e corr. p m. recenti (fuerat JA ty AF), Campanus, 
Zambertos. V in mg. habet: lari ds vj JB xaig AB, AF fisC^o- 
vs^aQa at BA, AF r^g BT ad tari lin. 20 relata. 

Eaolides, edd. Ueiberg et Menge. 4 




50 STOISEIfiN o'. 

IlavTOS ap« z^iydvov al Svo TikiVQal xfjs XotJt^ 
y,Ei%ovig dei xdvrri ^hzaka^^avo^Evai' onsQ eSei, 
Sit^ai. 



'Eav TQiymvov izl jinJg rtoi/ jrAeupiav axo 
lav iti^tttiDV 6vo iv&etai. ivTog 6vaTa&(a0tv, 
C 6vaza9£iaai zSv Aoiflrwv zov TQiydvov dvo 
Isvffav i2.dTT0vEs (liv ieovTat, (iBt^ova Sh 
avCav xsQLi^ovSiv. 

Tpiyiovov yap zoii ABF inl ^ias tAv xXsv^mv 
10 z^s Br «Jto ziov %BQ(tT(av zav B, F Svo av&stai iv- 
Tos (JwEfftfftraffav aC E^, JF' Xsym, ott aC Bz/, /JV 
zav Xoimmv tov T^iyt6vov Svo :tlevQmv rav BA, AF 
iXdaeovsg ftiv sleiv, fiei^ova 6s yaviav %SQii%ovaL zi^v 
v%o BJT Tijg vim BJr. 

zJirix&ai yaQ rj B^ inl z6 E. xal ixsl navTos 
zfftydvov aC 3vo itXsvpal r^g Xom^s fiBi^ovig sieiv, 
zot, ABE aqa zQtytovov aC 6vo nXsv^al at AB, AE 
zijs BE jisi^ovis staiv xoivij n^oaxsia&ia ^ EF' 
aC apa BA, AF tcov BE, EF fisiiovds eiatv. aa- 
20 Xtv, insl loiJ FE^ ZQtyiavov aC Svo nXsv^al aC FE, 
E/3 zfis V^ fi,sit,ovis siaiv, xotv^ ngoaxEie&a ^ jJB' 
aC FE, EB UQa tmv Tz/, z/B fisitovig Blatv. dXXa 
zav BE, EF fisii,ovEg iSEix&yieav aC BA, AF' JtoXX^ 
dfa aC BA, AF zmv BjJ. ^F ftsitovis Eiatv. 

XXI. Schol, ia Pappum III p. 1183,4. BoeUua p. 381,26. 

3. eldtv P. 4. jtliveiav 8vo tvfttiai vvatix^aaiv ivtot 

ano luv xegaTav ag^aiicvai al ProclnB. 6. Svq\ om. Pro- 

cIqb. 7. haxxovs F, Procloi. 8. niQif^ovai Proclus, Vbp. 
11. jr altvtui TM* P. 13. elai Vbp. nmtixoveiv PT. 



ELEMENTORUM LIBER I. 51 

Ergo in quouis triangulo duo latera reliquo ma- 
iora sunt quoquo modo coniuncta; quod erat demon- 
strandum. 

Si in uno latere trianguli a terminis duae rectae 
intus coniunguntur; rectae coniunctae reliquis duobus 
lateribus trianguli minores erunt^ maiorem autem an- 
gulum comprehendent. 

In triangulo enim ^JBF in uno latere JBF a ter- 

minis jB, F duae rectae intus coniungantur B^, ^F. 

dico, esse B^ + jr<BA + ^F et L B^r>BAr, 

educatur enim B^ 9A E. et quoniam in quouis 

-triangulo duo latera reliquo maiora sunt [prop. XXJ, 

in triangulo ABE erunt 

AB + AE > BE. com- 

munis adiiciatur£F. itaque 

^^ \ BA + Ar>BE+ Er 

[x.^vi/.4]. rursus quoniam 
^^in rEJ triangulo 

rE+ EJ> r^, 

communis adiiciatur jdB, itaque 

FE + E5 > Fz/ + JB. 
sed demonstratum est BA -{■ Ar> B£ + EF. ita- 
que multo magis BA + Ar> BJ + ^F. 




14. Bz/ri FdB F. 16. E] euan. F. 16. doiv\ PF; 

comp. b; noi unlgo. 17. Post nXBvaal in P del. r^g Xomriq 
pLsi. 18. siaiv] PF; comp. b; siai unlgo. nQoa- supra 

m. 2 b. ET] Br F. 19. slaiv] FP, comp. b; siai. uulgo. 

20. FEd] J add. m. 2 F. 21. siaiv] PFV; siai uulgo. 
^B] Bd b. 22. aga FE, EB F. 23. BA] corr. in AB 

V. 24. jr] AFF. siaiv] PF; siai uulgo. 

4* 



52 STOIXEIiiN a. 

IlaXiv, hcsl itKvtog Tptyrovou ij BXTog yovia t^jf 
evroq xal KJtevavTiow ^eitfov istiv, rou r^E aQU 
rptyoivot/ ^ ixTbg yavCK i] vno B^F ^ti^mv iiStI 
zije VTtb FE/J. Sta Tavra toCvvv xkX tov AB_E tqi- 
5 ymvov Tj ixtbs yavta r] vxo FEB jiHX^av ifftl r^g 
twro BAF. «kka r^s vnb FEB yLBCt,oiv iSeCxQ^rj ^ 
vno BdF' itolXa aga rj vno BAF (i,ei^av ierl f^g 
v«o BAV. 

'Eav ttfia tQiycovov inl fiiai tcov JiAcuptoi' ajto 

L(j rmv stBffdrwv Svo Ev&stai ivros Svata&m<iiv, at 0V' 

Sta&£tOcti twv Xoiamv xov rgiymvov dvo xkEvgmv 

ilatrovES ^lbv elaiv, fiBi^ova dl ycovCav ]tsQiBxovai.v 

oitEQ fSst dEl^ai. 

'?'■ 

\h 'Ex r(it(dv tv&Btav, a'i eioiv ieat rptoi taig 
So&ECSatg [iv&aiats], rQiymvov everiiaae&af 
det di rag dvo f^g Aotff^g ftditovag elvai jtdv- 
rfj ftETaKafi^avojidvas [dta rb xal attvxbg tq 
ydvov rag Svo aXevpag tijs Aotw^g liBi^ove 

SO elvai navtij (ietakaiL^ttvofiivas]- 

"Eeraeav ut So&Eieai tqeIs Ev&siai at A, B, P, 
atv al dvo r^g lomijg (lEt^oves eetmeav aavrri (tertt- 
Xan^avofiEvat, al (*iv A, B r^g F, at Si A, F Tijs B, 
xal ert ttl B, r r^g A' SsC Sij ix rtov iemv Taig A^ 

26 B, r TQiyavov eveti}ea09tti. 

'Exxtie&m Tts Ev^eta ij AE nEitBQac^ivvi [liv xarSt 

XXII. ProcluB p. lOB, 16. Entocitia in ApoUonium p. 10. 
BoetinB 1). 382, 1 (male), pattem demonstrationis habet Pro- 
clus p. 330 Bq. 

2. lyvot] it- in ras. b. hzCv] PF; laiC unlgo. TJE] j 
e oorr. F in. 2; mntat. io FEd V. op«] Bapra F, 



I 
I 



ELEMENTORUM LIBER I. 53 

rursus quoniam in quouis triangulo angulus extrin- 
secus positus maior est angulo interiore et opposito 
[prop. XVI], in triangulo rJE erit L B^r> FEJ. 
eadem de causa igitur etiam in triangulo j4BE erit 
irEB> BAF. uerum demonstratum est LBjdir> 
rSB. multo igitur magis BJr>BAr. 

Ergo si in uno latere trianguli a terminis duae 

rectae intus coniunguntur, rectae coniunctae reliquis 

duobas lateribus trianguli minores erunt, maiorem 

autem angulum comprehendent; quod erat demonstran- , 

dum. 

XXII. 

Ex tribus rectis, quae tribus datis aequales sunt, 
triangulum construere (oportet autem duas reliqua 
maiores esse quoquo modo coniunctas [prop. XX]). 

Sint tres datae rectae A^ By Fy quarum duae reliqua 
maiores sint quoquo modo coniunctae, A -^- B> Fy 
A -^- r> By B -];- r> A. oportet igitur ex rectis ae- 
qualibus rectis A, B, F triangulum construere. 

sumatur^) recta AE terminata in A, uersus E au- 



1) Proclum non ipsa uerba Euclidis citaref adparet. cfr. 
idem p. 102, 19. Augustum perperam post KAG p. 54, 5. sup- 
pleuisse: xal TSftvizmaav dU.^qXovg ot %vnXot, Tiata to K^ de- 
monstraui „Studien" p. 185. 

BJT^ J in ras. F. iGxiv PV. 4. TE.-/] eras. F. xavta] 
xa avta F; Tavra Ybp. 5. kaxCv P, ut lin. 7. 6. aXka 

ital T^ff P. 7. BdT] (alt) B^ in ras. suntV. ^ 12. bIoiv^ 
P; bIci uulgo. 15. aZ staiv tQial raig do&siaaig svQ-siaig taai 
ProcluB p. 329; sed p. 102: at sCaLV taai tQLol tatg Sod^eiaaig 
£v^£/at^. 16. svd^siaigli om. b; m. rec. P; supra p; mg. m. 2 V; 
om. EutoduB. 17. di] Proclus, Eutocius; d?} codd. xdg'] corr. 
ex xaig F. Svo] p i. 18 did xo — 20. fisxaXafipavoftivag] 
omnes codd., Boeiius; om. Proclus, Gampanus; contra Eutocius 
ea babuisse uidetur. 21. xQsig] om. p. 



54 ETOIXEIftN a. 

t'o A tinEiQos di xata ro E, xal xtied-0 t^ (ihv A 
iarj f] AZ, x^ ii\ B fOij 11 ZH, tfj d\ T /Oij 15 H&- 
xal x^vT^o} fihv rp Z, dttieT^^fiari di tto Z^ xvxXog 
yEyQttip&<o 6 dKA' itai.iv xsvtp^ (ihv ta H, SiaOx^- 
I (lati Se ta H& Kvxkos ysyQttfp&a o KA&, xal iitE- 
t,EV%&o}Gav ai KZ, KH' kiya, ori ix tQiwv EV&Eimv 
rmv Cstov tatg A, B, rzffiymvov awietaxai, xo KZH. 
'EtceI yaQ to Z 0r]iiEiov xivxQOv iexl xov ^KA 
xvxkov^ Cor} iaxlv ^ ZA Trj ZK' dXka tj Z^ t^ A 

10 iativ itJri. xal ^ KZ a^a xri A eGxiv i'et]. saAtv, 
Exel xb H oijfiECov xivxqov ioxl xov AK@ xvxkov, 
iat} iatlv ij H@ xij HK- dkka r/ H& x^ F ieti.v fmj- 
xtel ^ KH aqa tj} T ioxi.v Hoi]. iotl 6l xal 1] ZH 
zy B eat]' aC rpets «(>« Bv&slai al KZ, ZH, HKtqioX 

15 xats A, B,r isai Eloiv. 

'Ex tffiiav «pK £v&EtiDV xmv KZ, ZH, HK, ai eI- 
Oiv iOai, xffiol xtttg So&EtOats ev&Eiatg tals A, B, P, 
tgiyavov evvietaxat xo KZH' oxeq Edei notijoat. 



\ 



I npos T^ So&EtOjj BV&ei^ xal xa aQOS avxy 
et]jie£ci T^ tfo-S^ittfjj yavia sv&vyQdftna l'et]v • 
yavittv EV&vyQaii(iov Gvex^eaeftttt. 

XXm. BootiQB p. 382, 6. 

1. T^] pofltea hisertiiin m. 1 V. 2. ri\ (tert.) m.rec. P. 
3. (ifw] oni. b, Proulus. 4. nhI naltv V, Proclna. fii»] 

om. V, Procltiii. JMxaiijfmri 6i~\ *a\ diaef^iiati P, 7. avv- 
tffnjxe V; evvietatttt p. 10] corr. ei n3 b. 8. ydQ\ ovv 

V. texiv P. 9. ZJ] JZ F. aU F. ZJ] JZ V 

(ante J raa., Z utg. m. 2). 10. *al ^ KZ aea t^ A iexiv 

.'aij] mg. m. 2 V. 11. letCv Bb. AKe] KAB P, et in 

rft8.V. 12, aU'F. IS. KHl eorr. es K^ m. 2 P. 14. 
HK BF. IffTiv ro.)] mg. m. 2 V. Jotlv Sf P. 16. trav] 



K^^k 



J 



ELEMENTORUM LIBER I. 55 

tem infinita, et ponatur ^Z = A^ ZH= JJ, H@ = F. 

et centro Z radio autem Z/1 circulus describatur ^KA. 

rursus centro H radio autem H@ circulus describatur 

XA@f et ducantur KZ, KH, dico, ex tribus rectis 

aequalibus rectis A, By F triangulum constructum esse 

KZH. 

A 

B 
T 




nam quoniam Z punctum centrum est circuli AKAy 
erit ZA = ZK*^ uerum ZA = A] quare etiam KZ 
= A [«. Iw. 1].^) rursus quoniam H punctum cen- 
trum est circuli AK&^ erit H@ = HK] uerum H& 
= F; quare etiam KH=r. et praeterea ZH=B. 
itaque tres rectae KZ, ZHy HK tribus Ay B, F ae- 
quales sunt. 

Ergo ex tribus rectis KZ, ZH, HK, quae tribus 
datis rectis A, B^ F aequales sunt, triangulus con- 
structus est KZH] quod oportebat fieri. 

XXIII. 

Ad datam rectam et punctum in ea datum angu- 
lum rectilineum dato angulo rectilineo aequalem con- 
struere. 



1) Cfr. Alexander Aphrod. in anal. I fol. 8. Studien p. 196, 



Tov F. 17. TQiai] om. F. F] om. V. 18. tfvviataxat p. 
21. BvQ^Qaitfim ymvla Proclus. 



56 STOIXEISiN a. 

"Earut r/ (liv do&slCa sv&tia i] jiB^ ro fi\ mpos 
avrfi 0r][iEiov ro ^, 15 S^ So^tiOa yavla evfl-uyporfi- 
fios fj vxo ^FE' ^E? Si] ntQog tfj do&eiSrj Ev&sia rg 
AB xttl rra Kpog aik^ erj}ifia rra ^ t^ tfod. 
!> v^a Ev&vy^ttfifta xfi vno jdTE lurjv yavCav sv&v- 
yQaiifiov avSf^eae&ai. 

ElX^ip&m itp' ixKtdQttg zmv F^, FE tvxovra at}- 
(ir.ta TK ^, E, xai ixa^EVX^O) rj ^ E' xa\ ix rpirov 
fv&tiav, ai hleiv lOai. rptfft tatg F^, ^E, FE, rpt- 
10 yavov eweSTtttio ro AZH, Sszi fffijv tlvai zijv iikv 
rj zfj AZ, zi)v Se FE zrj AH, xal hi zijv JE rp 
ZH. 

'Eml ovv Svo aC JT, FE dvo tatg ZA, AH 

itiai eIsXv BxaziQa ixatdga, xal ^dsis V -^^ fidoei. zjj 

15 ZH leri, yavia «pa jj ijTo ATEyavia zfi tOTO ZAH 

iariv tOri. 

Hpog aga zf/ do&Eiatj Ev&sia rfj AB xal tp «goe 

avzfi erjfiEia rp A ztj Sa&eiajj yavia Ev&vypdn(ia) zy 

vxb AFE tarj ytovia Ev9-vYQafi[ioe avviatarai 17 vxo 

20 ZAH- oxEQ ISei noi^aai. 

xd'. 

'Eav 8vo rQiymva rag dvo itkEVQag Itatg'] Svo 

ai.BVQaig teag ixi] ExatE^av ixatipa, nyv 8h 

yaviav r^s yaviag fieitova ^'j;jj Trjv vno zmv 

26 teaiv evQeiidv aEpitxonEvrjv, xal tiiv §d 

fiaaEiog iiEi^ovtt E^Ei. 

Eera Svo zQiyava ra ABF, AEZ rag Svo Ttliv- 1 



1 



XXIV. Boetiua p, ; 




ELEMENTORUM LIBER I. 57 

Sit data recta j4B et punctum in ea datum j4 et 
datus angulus rectilineus ^FE. oportet igitur ad da- 
tam rectam ^J3 et punctum in ea datum j4 angulum 
rectilineum dato angulo rectilineo jdFE aequalem con- 

struere. 

sumantur in utraque Fz/, FE 

quaelibet puncta ^, E et ducatur 

^E, et ex . tribus rectis, quae ae- 

quales sunt tribus rectis Fz/, ^E, 

FEy triangulus construatur AZH, 

ita ut sit rd = AZ, TE = AH 

JE = ZH [prop. XXII]. 

iam quoniam duae rectae ^F, FE duabus ZA, 

AH aequales sunt altera alteri, et basis AE basi ZH 

aequalis, erit L ATE = ZAH [prop. VIII]. 

Ergo ad datam rectam ^J3 et punctum in ea da- 
tum A dato angulo rectilineo AFE aequalis con- 
structus est angulus rectilineus ZAH] quod oportebat 
fieri. 

XXIV. 

Si duo trianguli duo latera duobus lateribus ae- 
qualia habent alterum alteri et angulorum rectis ae- 
qualibus comprehensorum alterum altero maiorem ha- 
bent, etiam basim basi maiorem habebunt. 

Sint duo trianguli ABF, AEZ duo latera AB, 



add. V m. 2: xaig 8o^Blaaig sv^s^aig. xqiaCv P. FE] 

mutat. in £r V. 13. 8vo] (alt.) SvaC FB. ZA] AZ F. 
14. sxaxiQa'] Bnpra m. 1 F. 15. aga] m. 2 P. 19. avv- 

Cataxat p. 22. rag] om. Proclus. xaig] om. Proclus. 

Svo] (alt.) P, Proclus; dvaC uulgo. 23. ixV ^^ ^^*' ytovCav 

trjq ymvCas (isCiova xi^v Proclus. 



58 2T0IXEIHN tt'. 

Qccg r«g jiB, AF zatg 6vo aXsvQate raig ^E, ^Z 
taag ^iQvrK ixardQttv IxazdQa, i^v (tiv AB rij ^E 
f^v di AF T^ ^7., y\ Sh «Qog ta A yavia r^5 repog 
tra jd yavlaq \i£it,(nv ^ffrto* kiyia, ori xal ^daig ^ Bf' 
6 fideeaig t^g EZ (ici^cyv dsriv. 

'Exel yoQ fisi^iov rj vxo SAT yavCa «■^g itxh 
E^Z ymviag, urvEtfraito jrpog i^ ^E svQ^tia xal ra 
itffos avTfi SnjftEiai r^ iJ rfi vao BAF yaviif t(Si} 
vnh E^H, xal xfie&cj oxoztQa t(ov AT, ^Z fffii ij 

10 AH, xal iTtt^Evx&aaav aC EH, ZH. 

'Enal ovv l'e-1] dSTlv ^ jiev AB rfj AE^ ^ Sh A 
trj ^H, Svo Sij ai BA, AF Svel xatg Ed, AH iaat 
tialv ExaTSQtt ixttTEQtt' xal yavitt i] vnb^ BAF yetvCif 
%y vx6 EJH Csij- paSis «9« V SV ^dsu vfi EH 

15 istiv l'Si}. Ttdkiv, ixBi tsti istlv ij ^Z rrj AH, iST] 
iaxl xal i} vnb z/ffZ ymvitt rfi vTt.o ^ZH [lEi^mv 
«pa T) vito iiZH T^s vito EHZ' itoXlm UQa (LBit,mv 
iSTlv y\ vao EZH zijg vao EHZ, xal iitsl rptyto- 
vov isTi zo EZH ttEi^ova i%ov t^i' ti^ro EZH ym- 

BO viav T^g vno EHZ, imb 5f t^v (tetgova yaviav 
(Lsi^mv «ksvQa vMOTsivti, ftsi^mv UQtt xal nXsvQa i^ 
EH T^g EZ. rffjj S% ii EH t^ Sr- ntC^av aQa xal 
1) BF Tijs EZ. 

'Edv ttQtt Svo TQiymva ras Svo TiXsvQas Sval 

26 nXtvpatg CSttg ixV txarBQav ixaTiga, t^v Si yaviav 
T^S ycovias ftei^avtt ijifj t^v vnb rmv tSmv sv&siiav 
xsQLSxofiivnv, xa\ t^i/ ^dstv i^s ^dasmg (isi^ova t^si 
ontQ iSii Ssl%ai. 

1. 3va( BFV. 3. iS Si tieas tm A yBivla rijg nfig 

Tm d yoiii^reKl Pi yioria Si tj v-ao BAP jimWos t^s vni EdZ 
Thecra (BFVbp). 4. Iflioj] -oj in rafl. V. 6. ^ntC] el (iij 
B- (iE/£o>v] Pi liii^aiy iniv Theon (BFVbp), iiro BAT 



1 

I 

p 

V 

"M 

"i ■ 

I 

i 
i 



ELEMENTORUM LIBER I. 59 

^r duobus lateribus JE, /IZ aequalia habentes al- 
terum alteri, j4B^jdEetj^r=dZy et angulus ad 
^ positus maior sit angulo ad /d posito. dico, esse 
etiam Br> EZ, 

nam quoniam L BAF^ EJZy ad rectam JE et 
pimctum in ea positum /1 angulo BAF aequalis an- 
gulus EjdH construatur [prop. XXIII], et ponatur 
^H^Ar=JZy et ducantur EH, ZH. 

iam quoniam AB = ^E et Ar= JHy duae rec- 
tae BAj AF duabus EJy jdH aequales sunt altera 

alteri; et L BAT = EAH. ita- 
que Jjr = EH [prop. IV]. rur- 
sus quoniam AZ = /IHy erit 
etiam L ^HZ = AZH. itaque 
L JZH> EHZ [x. ivv, 8]. multo 
igitur magis L EZH>EHZ [id.]. 
et quoniam EZH triangulus est angulum EZH ma- 
iorem habens angulo EHZ, et sub maiore angulo maius 
latus subtendit [prop. XIX], erit etiam EH> EZ. 
uerum EH = BT. quare Br> EZ. 

Ergo si duo trianguU duo latera duobus lateribus 
aequalia habent alterum alteri et angulorum rectis 
aequalibus comprehensorum alterum altero maiorem 
habent, etiam basim basi maiorem habebunt; quod erat 
demonstrandum. 

ymvla T^g vno £z^Z ymvCug] BF pdaig t^? EZ pdaseag B. 8. 
avri] -ri in ras. V; avz6 P. 10. EH] PF; HE BVpb. 14. 
fen? laxCY. 16. -<:^Z]P; z^H BFVbp. z^H]P; z^Z BVbp 
et F corr. ex ^ Z m. 2. 16. iatCv P, ut lin. 19. xat] xal yonvCa 
Vp. zJHZ]JZHF. JZH] JHZ F. 19. ro EZH] eras. F. 
ytovCav] mg. m. 1 b. 20. EHZ] euan. F. 21. xa/J om. F. 
vXsvQd] eras. F. 22. riEH t^] mutat. in tiJ E if ^ V, id quod B 
habet. 24. raig dvaC Vp. 28. SsC^ai] noi^aai bp et V m. 1 
(corr. m. recens). 




rxorsEiiiN a'. 



'Eav diio tfiymva Tccg Svo xKbvq&£ dval 
nltVQttCg taag fitj ixnt^ffav Bxari^a, t^v Ss 
^detv tijs ^aeimg [lei^ova ixVi ""'^ ^V'" ytoviav-— 
5 T^g ytaviag ^ii^ova e^Ei t^w vxb rav iOav eu-9 
^Biav «EQitioy^ivTiv. 

"Eexta Svo TQiyaiva za ABF, ^EZ zccg Svo nlsv- 
pag tag jIB, jiF rat^ Svo nXevQatg taig jJE, ^Z 
fffag 1%°^" iKttxiqKV ixatiga, t^v ^ilv AB t^ iJE, 

10 TTiv Sl AF t^ AZ- pdaig Sh rj BV /SaVms tijs EZ 
[ici^iav iezw Xiyat, oTt xal yavia ij vnh BAF yaviag 
ri^S V7i6 EAZ fiBiieov itSriv 

Ei yaQ /*ij , ijTOi fei) iarlv avrfj ij iXaaa&}V lerj 
(tlv ovv avx latLV ^ vno BAF t;] VTto EAZ' iarj 

10 yap av ijv xal ^daig i) BF ^dati, Tfl EZ- ovx lati 
Si. ovx aqa tajj iarl yavia ij wjrfi BAFtf] iixb EAZ' 
ovSa (t^v iXdaaav iatlv t; imo BJF T^g viro 
EjdZ' ildeaav ya^ av t\v xal pdaig ij BT jSKffsog 
Ttjg EZ' ovx effTt Si' ovx apa ikdeaav iOTlv ij vno 

20 BAF yavia t% v7tb EAZ. iSiix&v Ss, ots ovSi 
(arf (tei^av aga ierlv i] vxo BAF Trjg vnb EAZ. 

Ettv apa Svo tpiycava Tag Svo nlfvpag Svel TtXsv- 
Qatg laag i^ij ixaTEQav ixatSQa, t^v S\ ^datv t^g ^^- 
ffECjg iisitovtt ixy, xttl T^v yiaviav trjg yavCag y,Bit,ova 

26 B^Bi rijv vnb rav tatov tv&Hmv iCBQiBxonivtjV osiep 
iSai SBtiat. 



XXV. BoetinB 



2. Tttg} ota. ProclQS. Svai] Svo Proclus; rarj Svoi 
3. tijv 8i paaivj K«i i^» puaiv ProclnB; tijf ^aeiv Si V. 
" ] om. P. 8. TBit Svel nlivgaSs] om p. Svei B 
itgat' p. 12. trjq vao] mg. m. 1 b. 



'EI,1 







ELEMENTORUM LIBER L 61 

XXV. 

Si duo trianguli duo latera duobus lateribus ae- 
qualia habent alterum alteri^ basiiu autem basi ma- 
iorem habent^ etiam angulorum rectis aequalibus com* 
prehensorum alterum altero maiorem habebunt. 

Sint duo trianguli JIBF, JEZ 
duo latera AB, AF duobus late- 
ribus JEy jdZ aequalia habentes 
alterum alteri^ AB => JE et 

Ar^ jz, 

basis autem BF maior sit basi 
EZ. dico, etiam esse LBAr> E^Z. 

nam si minus^ aut aequalis ei aut minor est. iam 
non est LSAr= E^Z, tum enim esset BF^^EZ 
[prop. IV]. sed non esi itaque non est L^^r = EAZ. 
neque uero est LBAF <iEdZ. tum enim esset 

Br<EZ [prop. XXIV]. 
sed non est. itaque non est L BAF <,E^Z^ ei de- 
monstratum est, ne aequalem quidem eum esse. quare 

LBAr>E^Z, 

Ergo si duo trianguli duo latera duobus lateribus 
aequalia habent alterum alteri, basim autem basi ma- 
iorem habent, etiam angulorum rectis aequalibus com- 
prehensorum alterum altero maiorem habebunt; quod 
erat demonstrandum. 

ovf] om. F. BAF yatvia Vp. 16. ^ ^aaiq Pp. iaxiv 

P. 16. tcriicxC^ Hari iaxCv PV; iaxCv tari p. 17 vno BAF 
ymvCcc V. 17. ov9i] ov V. iXccaaaiv] iXdxxoiv PBVbp. 
19. iaxiv P. iaxi di' ovx aQct] iaxtv oux F. 20. yatvCa] 
om. BFbp. ovd' Vbp. 21. BAF yatvCa Y, 22. dvaC] 
xa£g 9vaC F V, xatg dvo P. 26. r^f — nsQisxofiivriv] mg. m. 
1 P. T^v] tj sequente ras. 1 litt. F. 



62 ETOEiEIiiN a. 

'Eav dva Tpiyatva ras dvo ytaviaq Sval yo-- 
vittiq fdag 1x71 ixatd^av ixar^^tf xal fiiavnkev^] 
pKW fiia jtAewpa fSrjv ^rot rijv srpog zats ('ffatf^ 
> yoivittig ^ zfjv vjtozeivovOav vnh ^lav 
Itscyv ycavimv, xal zag i.OLaag xlevpagzatg Aoi-. 
Ttatg aXsvgats ^"«5 e^si [exardQav ixaztQ^i 
xttl tijv Xotnijv yaviav tij kotnfj y<ov 

"E6t(o Svo zQiyava za ABV, ^EZ r«g Svo ya>~ 

10 vias tag vno JBT, ETA dval zatg vao ^EZ, EZ^ 
ieag iiovza ixKTtpav ixariQa, t^r y,lv vno ABF r^ 
ujto ^EZ, ti)v Se V7t6 BFA zrj vno EZJ' i%ita) S^ 
xttl [liav aXsvQttP fita jtXevga i'aijv, nQorsQov zi^v rapos 
T«rg t<Sats yaviaig zrjv BF tfj EZ' Xiyco, ozi xaX zag 

16 Xotnag nlevQag tatg XoiTCaig nXevQatg teag i^et ixa- 
ziQav ixatiQtt, zijv ftiv AB ry ^E z^^v Si AF tg 
^Z, xttX t^v Xoixr}v yaviav t^ Xouf^ yavi^, f^v i 
BAF rfj vith EdZ. 

Ei yctQ aviSos iotiv ij AB zfj ^E, (lia avrav ^C^m 

20 Itov iaziv. eazc3 (lei^av rj AB, xal x&Ca^a zfi AEtO^ 
5j BH, xaX inetevxS^co y HV. 

'EmX ovv tet/ iazXv ^ (liv BH zfj AE, 17 Si SM 
Tp EZ, Svo 3f) ttl BH, Br SvaX ttttg dE, EZ Csat^ 
elalv exttreQtt ixateQa' xa\ ytovia ^ vao HBT ymvia 

26 zfi vno AEZ lar) iariV /5aO(s «pa rj HF ^uasi zfj 
AZ tOfi iotiv, xal ro HBF zQiyovov zip ^EZ zQt- 






XKVI. Olympiod. in meteorol. II p. 110. Boetius p. 382, Ij. i 



2. loe] om. ProcIuH. Svai] Sva ProciuB; tait Bvai V, 

OlympiodoruB. 3. ^ai] Sx^ Si %al Proclus. 7. Htnifav 

fMKT^Sa] om. Proclaa; cfr, p. 66,15. 8. yiovlif tvriv i^ti " 



rik 



J 



ELEMENTORUM LIBER I. 63 

XXVI. 

Si duo trianguli duos angulos duobus angulis ae- 
quales habent alterum alteri et unum latus uni lateri 
aequale^ siue quod ad angulos aequales positum est^ 
siue quod sub altero angulorum aequalium subtendit, 
etiam reliqua latera reliquis lateribus aequalia habe- 
bunt alterum alteri et reliquum angulum reliquo an- 
gulo. 

Sint duo trianguli ABF^ JEZ duos angulos ABF, 
BFA duobus JEZy EZ/1 aequales habentes alterum 
alteri, L ABT = ^EZ etLBrA = EZJ, et habeant 

etiam unum latus uni lateri 
aequale, prius quod ad an- 
gulos aequales positum est, 
BF^EZ. dico, etiam reli- 
qua latera reliquis lateribus 
aequalia eos habituros esse 
alterum alteri, AB = JE ei Ar= JZ^ et reliquum 
angulum reliquo angulo, LBAF = EAZ, 

nam si ^jB lateri ^E inaequale est, alterutrum 
eorum maius est. sit maius AB, et ponatur BH = 
^Ej et ducatur HF. 

iam quoniam BH= JE et Jjr= EZ, duae rec- 
tae BHy BF duabus AE^ EZ aequales sunt altera 
alteri; et L HBr=^EZ. itaque Hr=AZ et 
A HBr= dEZy et reliqui anguli reliquisaequales erunt, 

Proclus, Boetius. (non Olympiodorus). 9. iatmaav V. 11. 
T^] corr. ex t^v m. rec. P, ut lin. 12. 12. vjrdj (alt.) m. 2 b. 
13. nXBV(^a\ supra m. 1 p. 15. xaig Xomaiq nXsvQaig Tag 

Xotnag nXsvQag F. 20. iaxiv] iaxai V. 21. BHl PB; HB 
FVbp. Post Instsvx^m ras. 4 litt. p. 26. htCvl^ PF; 

comp. b; lctC uulgo. 26. icxlv] PF; hxC uulgo. HBF] 
PB; HTB FVbp. 




64 STOIXEIIJN «'. 

ymvq} faov ietiv, x«l aC AotiiKt yaviai tats Xoiiratg 
yiaviais ^1«' Eflovrat, vip' ag aC Csai itXsvQal vizo- 
xsivoveiv fffij cpa ^ wito HFB ■yavia rjj ureo A2.K. 
kAAk ^ ujio JZE T^ iJno Efji vnoxEitai fOr}' xal 
6 ^ vffo BFH aga rfj vtco BFA iST) iotiv, vf ildeoiav 
Tfi [isi^ovf uxiQ dSvvaTOV. ovx apa avtSos b0tiv 
AB tfi JE. iGi} uga. lati di xat ij BF rfj EZ fst) 
Svo Sri at AB, BT Sval zaig ^E, EZ taai aislv 
ixatiifa ixaziQa' xal ymvia rj vab ABryavta ijj vao 

10 ^EZ iativ iai}' /Sitffts «pa ^ AF ^dati tfj jJZ lOn 
iativ, xal i.out7i yatvia ^ vnb BAT ty Aoiraj) ymvia 
T^ ijjro EjdZ iaij ietiv, 

'AkXa Sij Tcdhv iaraeav aC uiro ras /'ffag yaviag 
jckevQal vnoTeivoveat teai, as ^ AB t^ AE' kiysa 

15 Ttdi.tv, oTi xal at KontaX ^lavQal tatg AoLnatg itXsvQals 
ieai itJavrai, tj fiiv AF ty ^dZ, ij di BF xfj EZ 
xal ht fi }.oiJti] ymvia rj vnb BAF tfj AotMJJ ymvi^ 
tfj V7tb EJZ leri iativ. 

Ei yaQ avteos iattv i) BF r^ EZ^ y.La ixinav 

20 ^ei^mv ieTtv. tatm fisi^mv, ai Svvazov, i} BF, xal 
xBie%m tfi EZ iari t} B&, xal inetevx^''^ V ^®- ^'^^ 
ixtl ier\ iexlv rj fiJv B& xfj EZ jf 6i AB rp z/B, 
Svo S^ aC AB, B@ dval ratg JE, EZ taai EtaXv 
ExariQa ixatiQif xal ymvias ieag mQiixovetv ^deig 

26 aQtt 7] A& pdeti tfj ^Z foj) iaxiv, xal xb AB0 tqC- 
j-ravov xp AEZ tptymvm Haov ietiv. xal af. kotital 
ymviat zats ^otJiarg yatviats tacu iaovzai, v(p' ag at 
iaat TtXevQal vMxtiJoveiv lat] aga iazlv 17 ujto B&A 
ytavia xy vxb EZ^, dkXa 7} uiro EZ^d ttj vxb BFA 

1. ietiv} PF; comp. bp; iatC B; larai V. 2. faovrtu 

i%atig-x fxoriey V. 4. 15] aupra V. JZEJ .JEZ Fj 



1 



1 






I elementor™ liber I. 65 

fenb quibus aequalia latera subtendunt [prop. IV]. 
quare LHrB = AZE. ueram i^ZE = BTJ, ut 
aupposuimus. ergo etiam LBVH = BFA [x. ivv. 1], 
minor maJori [x. Ivv. 8]; quod fieri non potest, ita- 
qoe jiB lateri z/E inaequale non est. aequale igitur. 
nerum etiam BF = EZ. duae rectae igitur AB, 
Br duabuB ^E, EZ aequales suut altera alteri; et L 
JBr= ^EZ. quare ^r= ^Z et LBJT^' E^Z 
[prop. IVJ. 

lam rursus latera sub aequaiibus angiilis sub- 
tendentia') aequalia sint, uelut AB = /iE. dico rur- 
etiam reliqua latera reliquis lateribus aequalia 
ire, Ar= dZ et BT = EZ, et praeterea reliquum 
.lum BAF reliquo angulo E^Z aequalem esqe 

nam sifiriateri £Z inaequale est, alterutram eorum 
maiuaest. sitmaius, si fieri potest, Bf, etponaturSi9 = 
EZ, et ducatur -^®. etqtioniamB® = £Z etAB=iJE, 
duaerectae vJB,£@duabus.J£, EZaequalessuutaltera 
alteri. et aequales augulos comprehendunt. itaque A& 
-=^Z etA^B0 = ^£Z, et reliqui anguli reliquis 
angulis aequales erunt, sub quibus aequalia latera sub- 
tendnnt. quare LB&A=EZ^. uerum LEZJ = BrA. 
1) A! et itts lin. 13 abesse debebant. 

■eorr. m. S. BrA] corr. ei EFd m. 1 b. 6. BTA] corr. 

mp. jrS F. 7, aga. l<rci] uea iativ. Imtr P. 8. Svai B. 

W^IO. JEZ\ corr. ex AZ m.2 b. 11. Inlr] PF; tati uolgo. 

■^ ij lotKT) F et V m.2. ^ B-*r] rAE P. rn loiw^l loiwj 
V; coiT. m, 2. 13. dXla 5^] bis b, Bemel punctis uel. m. 

leceDB. 17. Tiaf\ e corr. V. r^] om. b; poatea inaertum 

V. ymvi^} om. b. 20. ct Svvcitov (tf/Joiv Theon? (BPV 

bp), tl\ add. ra. receuti b. ^ BF ins EZ P. 34. jtrpi- 
^mimv] PBF; nepiEjovat nulgo. 25. ^oiC»'] PF; ^oi^ uuIbo. 
26. ^in^*>] PF; coiap, p; itfi/ nulgo. 27. imvTat ixaztfia 

inteieea V. 29. dV.' P. ij] postea add. m. 1 P. 

. Eoslidei, «dd.BtibeTB cl Upnge. 5 ^ 



66 STOIXEIiiN a: 

iativ fffij" TQiycavov S!] rov A®r rj ixTog ymvia tj 
v^o B®J tojj i<STl z-fj ivtog xal a^evavTiov t^ vno 
BFA' ojtep aSvvuTov. ovx &Qa aviGoq iaziv ij BP 
T^ EZ' Htst} «p«. istl di xal 7j AB Tfj ^E ftSfj. 6vo 

5 Si} aC AB, Br Svo Talq iJE, EZ tcai eielv exatipa 
ixatiga ' xal yavtag ('Ocg Jifpt^jjowfft " ^dffig aQtt ^ 
AF pdesi trj z/Z tari iOTiv, xal ro ABF tQiymvov 
Tp i^EZ irpiyojvp taov Kttl Aots^ ymvia i} vitb BAV 
ffj komfj ytavitt t^ hno EA7. fffij. 

1 'Eav tt^a Svo TQiyavtt tttg Svo yoaviag Svdl 
yavittis teas i'xjl ixatiffav ixatsQa xal fiiav nlsv- 
gav [tta aXsvQtt fei^v tjtot ttiv nQog Tatg ieaig yw- 
vittiq, ^ riiv vnotsivovaav vno (liav tmv lemv yavimv, 
xal tag loixag nXEVQag tatg loizatg zlevQatg ttSag 
15 €^£i xal Tjjr ioinrjv ymviav tfi Aoijr^ yatvia' offfp idEt 
Sstlai. 

xs;. 

'Eav tig Svo tvQ^Eiag Ev&sitt iiLxintovatt tas 
ivttkka% yaviag tffKg «^iijAKtg Ttoijj, xaQii^XTi- 
20 Xoi Seovttti alK^kttis at Ev&etai. 

Eiq yaff dvo ev&tittg tag AB, Fd sv&Eta i^xi- 
movea ^ EZ tag ivaXXa^ ymviag tag vjto AEZ, EZ^ 
i'eag aXXriXatg jrotiiro' Xiym, oTi. aa^dXXijXog ieTiv fj 

AB Tfj r^. 

!, Ei yaif fi^, ixpaKKoitevat ai AB, Fjd avy,nEeovv- 
tai ^TOi inX ta B, /1 ft^PI ? i^ti r« A, T. ixptfiX'^- 



XX VII, Philop. i 



Boetiua p. 3 



1. Poflt firij Theon n.dd. xal q iiri BBJ optf 1^ ino SFA 
lonv faij (BFVbpi in F aga Bupra aor. et pro SFA legitur 
Srj); eadem P mg. manu rec. B. imiv P, ut lin. 4. 5. 
8v}{ BPp. 7. iexiv] PF; ieii uulgo. B. taov iatl Theoo 



I 



■ ELEMENTORIIM LUiER 1, 67 

Haque in triangulo A&F angulua extrinsecus positus 
B&j4 aequalia est angulo interiori et opposito BFA; 
quod fieri non potest [prop. XVI]. quare BF lateri 
EZ inaequale non est; aequale igitur. uerum etiam 
AB = ^E. itaque duae rectae AB, BP duabus ^E, 
EZ aequales suot altera alteri. et angulus aequales 
compreheudunt, itaque basis AV basi jJZ aequalis 
eat, et triangulus ABF triangulo ^EZ aequalis, et 
reliquus angulus BAT reliquo angulo Ei3Z aequalia. 
Ergo si duo trianguli duos angulos duobus angulis 
aequales habent alterum alteri et unum latus uni la- 
teri aequale, siue quod ad augulos aequales positum 
eat, siue quod sub altero augulorum aequalium sub- 
teodit, etiam reliqua latera reliquis lateribus aequalia 
babebunt et reliquum angulum reliquo angulo; quoil 
^t demoustrandum. 

XXVII. 

Si recta in duas rectas incidens alteruos angulos 
inter se aequales effeceritj rectae inter se parallelae 
erunt. 

Nam in duas rectas AB, FJ recta incidens EZ 
Bngulos altemos AEZ, EZ/f inier se aequales efficiat. 
dico, AB rectae T^ parallelam esse. 

nam si minus, A&, TJ productae coocurrent aut 
ad partes B, ^ aut ad A, T partea, producantur et 

(BTbp; t<sov iatlv F); laii om. P. Io.tij] P, V m. 1; J 

loiw^ BF, V m 2, bp; cfr, p. 64,11. 9. 1^] Bupra ni, 2 v. 

fiJij ietiv BFbp. 10. a^a]^ BUpra m. 1 P. infe Svai 

BVp 11. Ante %ai m. recenti add.Vr ^m 3i. 14. tcIsv- 

gas] i» raa. m. 1 P. 15. yiavi^] comp. inaert. V. 16, ieC- 
iortj ra«. p. 18. iimieoiaa F '{™pra m. 1: yp. IfiaijiTOveo). 

»0. at} om. V, 24. r^ tv»tia V. 



1 



68 STOIXElftN a\ 1 

e&taOav xal OvfutiTttdtasav ixl ra B,^ n^Qrj xKza ro 
H. TQiyavov dij zov HEZ ^ ixzbs yavCa ^ vao AEZ 
tet} iatl ty ivtds xal a«EvavtCov t^ vito EZH' onsp 
i«zlv iSvvatoV ovx «pa «f ^B, F^ ix§ali.6nEvai 
5 SviiTtteovvtai iitl rH B, /1 iti(fr]. ojioimg S^ S£tj^&'^- 
fffTcti, Qzt ovSh inl tu A, F' at Si ixl nijSare^a tu 
^ep?! UvfiitiTttovsat jtaealXijXoC siStV wapaAAjjAos «pa 
iexlv it AB zrt F^. 

'Eav «pa £^g Svo ev&aias av&tta ifizCjtzovea tas 
10 ivalXa^ yavCag fuas all^^iais aoif/, jrapaAAijAoi i^ffov- 
rK( aC sv&Btaf ojttQ ^Sei Sai^ai. 



'Eav eis dvo ev&sCas ev&aZa ifimCnrovaa r^v 

ixzos yatviav zf/ ivtos xal aasvavrCov xal iicl 

15 Ta avta (ii^ri Herjv «ot^ 7} tas ivzbs xal inl za 

rtvta itiffr} SvaXv opfraJg iflKg, ««paAAijAi 

zat all'^i.uis al ev&etai. 

Eis yaQ Svo ev9eias tas AB, FA ev&aCa i^tjtC— 

sttovea 17 EZ rijv ixtos yaviav rijv wreo EHB rj; iv 

20 Tog xal aitevavtiov yetvi^ zjj vao H&A tetfv noisCzoi 

7} ricg ivrog xal inl tit avra (i/pi) riij iiao SH&, 

XSVIK. BoetinB p- 382, 26. 



i 



2. Poet H add. arjittiot (comp.) Y man. recenti. ^ IxTogl 
— AEZ} mg. m. 1 P. 3. i-oTj] raa. FY (f»ett'"' OrynaeuB, fis^^ 
£bw GregonuH). ffftl'»' P. 1^] «le TT, Grynaeua, 

uxevavtlov'] EXEvixvyraVKE v, praeterea. ymr^ae (oomp.) mg. m. 2 
F; m. 1 Bini! dnbio foit «Mevttvtiov. In V poBt hoo verbum 
■fmvCas (comp.) inseruit m. recenH.; yaivCas hab. Grynaeua. 
iS] tjjs FV". u«d] om. F. Poflt EZH in F. m. 2 et inV 
m. reeentiasima adil. dlia xcel foq, quod babet Gryuaeus. acrip- 
turam receptaini habeut PBbp, Campaiiue, Zambectns, alt«r 
codex Grjnaei, 4. lazCv] om. p. 5, *)j] 3i P. 6. ovd' p. 



a 



ELEMENTORUM LIBER I. 69 

concarrant ad A, ^ partes in puncto H. in triangolo 
igitur HEZ angulus extrinsecus positus jiEZ aequalis 




est angulo interiori et opposito EZH\ quod fieri non 
potest [prop. XVI]. quare AB^ F^ rectae productae 
non concurrent ad A, ^ partes. similiter demonstra- 
bimus, eas ne ad^, Fquidem partes concurrere; quae 
autem ad neutras partes concurrunt^ parallelae sunt 
[de£ 23]. itaque AB rectae FA parallela est 

Ergo si recta in duas rectas incidens altemos an- 
gulos inter se aequales effecerity rectae inter se paral- 
lelae erunt; quod erat demonstrandum. 

xxvm. 

Si recta in duas rectas iucidens angulum exteri- 
orem interiori et opposito et ad easdem partes sito 
angulo aequalem effecerit aut angulos interiores et ad 
easdem partes sitos duobus rectis aequales^ parallelae 
inter se erunt rectae. 

nam recta EZ in duas rectas AB^ Fz/ incidens 
angulum exteriorem EHB angulo interiori et oppo- 
sito HQ^ aequalem efficiat aut angulos interiores et 

^«] 9' Pp. 7. bUsiv] PF; bIci uulgo. 9. «&] supra 

m. 2 y. 11. at] om. b; eras. F. 15. Post kvxoq 

add. V m. 2 ytoviag (comp.). %aC] supra m. 2 V. 16. 

dvciv] Svo Produs. 17. aXXiiXaLg] om. Proclus. at] om. 

Y, Proclns. 20. insvavtCov 9, dnsvavxCag p. Post dn- 

svavxiov add. F: ymvCa (m. recenti) xal Inl ra avta ii>i(ffi; cfr. 
Campanus. ymvia] om. BFp. 21. Post /ttf^i; m. 2 FV 

add. xd BJ, 



70 STOIXEIftN a, 

H0jJ dvalv OQ^aig [eag- klya, oti 3nr(iaAAi;Ao'g iertv J 
i\ AB r^ rj. 

'E%il ya^ tStj iffrlv ii wro EHB r^ vtco HS^, I 

a3ilee i] intb EHB t^ vno AH& ietiv tgi], xal ^ [ 

5 imo AH& aga r^ V7td H&A ietiv fuij- xa{ eiaiv ■ 

ivaXka^' ffapaiijjAos aga iozlv i} A B rij Tz/. 

ndkiv, i%t\ at vxo BH@, H@A Svo o 
ttttti sleCv, eial d\ x«l at vno AH&, BH@ Svalv 1 
Off&ats ieai, tti aga wjto AH&, BH& targ wa^ ] 
10 BH@, H&A tatti daiv xoivij atp^^^eS^to ^ imo BH&' 
lotnii apa 17 vxo AH& lotaij r^ vitb H&A iariv \ 
taij' xai siaiv ivakla^' TtaQalk^Xos ctQa iatlv ^ AS \ 

Tfi rj. 

'Eav ttQtt cig Svo dv&etas tv&Bia ifimittovea t^ ' 
IS ixrbg yavittv ty ivrog xal aTtBvavtiov xa\ iiti ra avtu 
(ligr} rffijv Jtoi^ i] TKff ivtbg xal inl ta avta /ifpi/ 
Svelv OQ&atg taag, xaQdllfiXot leovrai at ei&stai' 

20 'H sig rag itKpaAAijAous BV&Biag Bv&Bia ift- 
Ttintovea tdg te ivaXktt\ ytaviag teag ttXX-^kati 
Ttoiil xaX ti\v ixrog rrj ivtbg xa\ axsvavrioi 
iaijv xa\ rag ivrbg xal inl za avtd [liQrj dvelv 
OQ&atg taag. 

26 Els yag TtaQakXiiXovi svStiag zag AB, fz/ tv&tta | 

3. Poat EHB in V add. yoivi'B m. 2 (comp.), H8J] 

HB J F, Bfld B e corr, 4. tari htlv p. 5. Ante H&^ 

raa, 1 litt. F. hii lativ p. T. Svaiv Bp. S. slaiv Caai 
p. ttalv ie P. af] Bnpra m. 1 b. 9. at Sea] aga at F. 

10. tlaiv] PBF, comp. bj slai Diilgo. II. fiiTi lati 

la. lativ] om. P, AB] e corr. F; in raa. b. 16. a 
ilea p. 21. rt] om. F, eupra m. aV. yavias] om. Proclui. 

all^Xats] om. ProeiuB. 22. «oie»] oorr. ex. notg V. " ' 



1 
i 



'•• r M 

Procloi. ^l 



ELEMENTORUM UBER I. 71 

ad easdem partes sitos BH&j H&^ duobus rectis 
aequales. dico^ parallelam esse j4B rectae JTz/. 

nam quoniam L EHB = H0^ et L EHB^AHB 
[prop. XV], erit etiam AH& = H&A [x. iw. 1]. et 
sunt alterni. itaque AB parallela est rectae FA 
[prop. XXVII]. 

rursus quoniam BH& -f' H&d duobus rectis ae- 
quales sunt, et etiam AH& -{- ^HB duobus rectis 

aequales [prop.XIU], erunt etiam 
AH& + BH& — BH& + H&A 
[x.ivvA]. subtrahatur, qui com- 
munis est L BH&. itaque 

L AH& = H&A [x. iw. 3]. 
et sunt altemi. itaque AB par- 
allela est rectae FA [prop.XXVII]* 
Ergo si recta in duas rectas incidens angulum ex* 
teriorem interiori et opposito et ad easdem partes sito 
angulo aequalem effecerit aut angulos interiores et ad 
easdem partes sitos duobus rectis aequales, parallelae 
inter se erunt rectae; quod erat demonstrandum. 

XXIX. 

Recta in rectas parallelas incidens et angulos al- 
ternos inter se aequales efficit et angulum exteriorem 
interiori et opposito aequalem et interiores ad easdem- 
que partes sitos duobus rectis aequales. 

nam in rectas parallelas AB, FA recta incidat 




XXIX. Boetius p. 383, 1. 



ansvavxiov — 23. iyrd?] apud Proclum exciderunt. dnsvav- 
xCaq p. 23. tcriv'] P, Campanus; xal ^nl xa avTa fikiqTi lar^v 
Theon (BFVbp, Boetius). dvciv^ dvo Proclus. 



ETOIXElliN a 



1 



ifinimiza ij EZ ■ Xiyco, ott. tas ivak^a^ yaviag tag 
vao AH®, H&J fffKS tout xal r^i/ ixrbg ycoviav 
Ttjv wwo EHB Tjj ivros xal aTttvavxCov ztj vito H®/i 
iCviv xai taq ivTog xa! inl ra avra HEQtj tag vx6' 
r< BH0, H&J Svalv opdaCs Caag. 

Ei yap avt<s6g ieztv ^ wwd AH® zij vjto H®^, 
(ttK avTmv iisi^av ifftiv. efftra ft£/£rav 7} vxo AH&' 
xoivi] jCQoOy.Bie&a ^ vitb BH&' at aga vitb AH&, 
BH® Tcov vitb BH@, H®^ [isClovig eiGiv. «AA« uC 

10 U3EO AH@, BH® 6vesiv op&atg iaai slaiv. [xal] at 
apa V7tb BH&, H&J Svo oQ&av ikdseovig eiei.v. aC 
Sh Kit' iiaseovav ^ ffvo op&mv ix^alloiievai sig aitet- 
pov evfinLazovaiv' ai aga AB, Pjd ixfiaXi.oitEvai sts 
antiQov Ov[ateaovvtai' ov avfiTtijttovai Si Sia zb wap- 

15 aAAijAovg avtag vitoxsta&ai' ovx aga Svtaos iativ ^ 
vxb AH&zfi VJtb H@^ ■ lar} UQa. akla t) vnb AH& 
tfi vnb EHB iattv iajj' xal ^ vab EHB apa t^ 
vTtb H®ji iaziv tet). xotvi} apoaxtie&a {j vjtb BH®' 
al apa VTtb EHS, BH& taig vnb BH&, H&^ isat 

20 sieiv. alXa aC vitb EHB, BH® Svo oQ^utg taat 
siaiv xal at vxb BH&, H®^ apa Svo opd-atg latti, 
tiaiv. 

'H affa sig tag naffall^^lovg sv&eias ev&eta ift- 
jiiatovaa zag te ivai.i.tt^ ycovias tOag dklijXaig irotEf' 

25 xal tijv ixtbs tij ivtag xal, ansvavtiov fffijw ical 

1. zag] PP et V m.li tB« re Bbp et V m. 3. S. 

i*o*T('as p. tf/] P; «"^ ^"' '« """ f"'ei ^V Theoa (BFVi 

bpl, Campanua. H»J] H aupra sct. m. 1 F. 4. fiDj V.# 
7. iatiF. AH&] FVb; AH9 x^e vwh HS^ P; AH9. *ai\ 
inil iiiifov {axlv ^ iffo AH9 t^s ino H&.J Bp, et mg, m.l I 
V, 9. dll' F. 10. BHG] SHB B et e corr. V. tMM 

V, comp. b. MttfT] ora. P. 12. «t.'] jji' b. 

jrftiMVOiv — H. aneipo»] om, p. 16. xj] i^s B. 



I 



"V 



ELEMENTOEUM LIBER I. 73 

EZ. dico, ean] angulos alteruos AH&, H&zl aequales 
e£Gcere et angulum exteriorem EHB interiori et op- 
posito H&^ aequalem et interiores ad easdemque 
partes sitos BH&, H®^ duobus rectis aequales, 

nam si i AH& angulo H&zl inaequalia est, alter- 
uter eorum maior est. sit i AH& maior. eommuniB 
adiiciatur i BH&. itaque 
AH& + BH& > BH& -\-H®A 
[xMv.^}. uemm AH& + BH® 
duobus rectis aequales sunt [prop. 
XIII}. quare BH& + H&^ du- 
obus rectia minores eunt. quae 
autem ex angulis minoribus, 
quam sunt duo recti, producuntur rectae in infinitum, 
concurrent [ttiV. 5]. itaque AB, F^ productae in in- 
finitum concurrent. uerum non eoncurrunt, quja sup- 
ponuntur parallelae. quare i AH& angulo H&/J 
ioaequalis non est. aequalis igitur. 

sed L AH& = EHB [prop. XTj. quare etiam 
i, EHB == H&d [x. fvv. 1]. communis adiiciatur 
L BH&. itaque /. EHB + BH® = BH& + H&^ 
[x. ^w. 2]. uerum EHB -f Bn& duobus rectis aequalea 
8unt [prop. XIII]. quare etiam BH®-\-H®^ duobus 
rectis aequales sunt. 

Ei^o recta in rectaa paralletas incidesa et anguloa 
alternos inter se aequales efficit et angulum exte- 
riorem angulo interiori et opposito aequaleni et inte- 

litt. H0 in ran. F. dXla] dH' F. 19, vxo] (prius) at vx6 b. 

BHe. H»^] H bia e cotr. V. 20. dW F. ivalv Bp. 
21. tiaiir] PBF; ffoiTuulgo. Sveh PBp. tl<iiv f<t«( BF. 

. ^j e corr. V. 24. rtl om. P. 25. Iws rg] m, 2 F. 
ftrq»] om. Pj %ciliiiixiiiivtiiii(iiriiijvBFV\i 



ETOISEIliN 



iv%6s xal inl rd avTa f^Ep») dvalv o^Q^ats ttsag' onsff 
ISet Sst^ai. 

AC rij KVt^ tv&eCa naQaXXrjXot xal aXXijXaig 
6 iitfl %aQaXli]Xot. 

"EffEo ixatipa rav AB, V^ r^ EZ na^dXXijXog' 
Xiyfo, oti xal tj AB t^ F^ eoti maQaXXrjXoS' 

'EfimTPriTm y«p sig aut«s fufl^fia ^ HK. 

Kal sael dq Xa^aXX-riXovg ^v&eiag rag AB, EZ 
10 £v&£ta hfinintaxev i] HK, ierj a^a ij vno AHK t^ 
vito H&Z. TtaXi.v, mel eig napaXXriXovg ev&Eiag tkg 
EZ, r^ ev&eta ejiaintaixev ij HK, tts-q ketlv fj imo 
H&Z f^ vito HKJ. edeix&ti 61 xal rj v%6 AHK 
ty uffo H®Z fffjj. xal t'i vao AHK apa rrj vao 
15 HKA ievtv ierj ■ xai eletv evaXXa%. TtaQalXtiXog a^a 
effilv 7) AB rij Fzi. 

[AC apa rfi avt^ ev&eia naQttXlijXoi xal aXX^^Xaig 
eial aaQaXXTiXof] ojTEp iSet det^at. 



^ia toO do&ivTog Ori^eCov xy So%eCe\i ew 
&eC« jiaQaXXrjXov ev&elav yQanfi^v ayay 

"Eatta 10 jtiv 6o^%v ei\^elov t6 A^ y\ S\ So^ttet^ 
iv&eta i] BF- Set Sri dia roiJ A armeiov trj BV ei-^ 
&eta jtaifaXXrjXov ev&etav ytfafiiiriv dyayetv. 

XXX. BoetiuB p. 383, B. XXXI. Boetiua p.383, 7. 



1 



1. hxos xo/J om. P. 6. AB] JE >p. 7. ioTiv _. 
9. «tt^— 10. HK] mg, m. 1 P. 11, els] tCs lueV. tvlfeiai 
Svo tv&iias P. 12. iivaixTanev] in rae. PF; dein add. xoivi) 
F. ;jj (alt.) corr. ei: x^ P. 13. HKJ] cotr. ei 0KJ m. 
rec. P. 14. opo] BUpm comp. m. I b. 16. @KJ P, eorr, 
m. rec. 16. icTiv] om. F. AB] intei vl et fi rae. 1 UtL 




ELEMENTOBUM LIBEB L 75 

riorea ad easdemque partes sitos duobus rectis aequa- 
les; quod erat demonstrandum. 

XXX. 

Quae eidem rectae parallelae sunt, etiam inter se 
parallelae suni 

sit utraque jiB, F^ rectae EZ parallela. dico, 
etiam AB rectae Fd parallelam esse. 

nam in eas incidat recta HK. et quoniam in 

rectas parallelas AB, EZ recta 

A. :^ Jl incidit HKy erit 

J O ^ Z L AHK = H&Z [prop. XXIX]. 

rursus quoniam in rectas paral- 
■^ lelas EZ, FA recta incidit HK, 
erit L H0Z = HKA [prop. 
XXIX]. sed demonstratum est^ esse etiam 

L AHK -= HeZ. 
quare etiam L AHK = HKA [x. ivv. 1]. et sunt al- 
temL itaque^B rectaeFz/ parallela est [prop. XX VII] ; 
quod erat demonstrandum. 

. XXXI. 

Per datum punctum datae rectae parallelam rectam 

lineam ducere. 

Sit datum punctum A, data 

■^ ' ^ autem recta BF. oportet igitur 

^ per A punctum rectae BF par- 

^ ^ allelam rectam lineam ducere. 

F. T^] T^ff b. 17. at uga — 18. nocQaXXTiXot] om. PBbp; 
mg. m. 2 PV. 17. a^a] .om. F V. 20. Post nri^dov in P 
add. 8 ^1} icxw iicl avv^s; del. m. 1; similiter Campanus; sed 
Proclus non habuit p. 876, 5 sqq. 



76 



ETOIXEIGN c 



EiX^ip&o inl i^g Br tvxov oij/wfov i 
inE^tvx^'^ V -^'^' "''i OwfeTttTm irpog t^ jdA tvi 
xal rp nQog avTrj 0T](ieia r^p A r^ v%o AJF ymv&^ 
aari 11 vni) JAE- kuI Bxpi^X^a&a iiC iv&eias ' 

(• EA iv»Ei:a Tj AZ. 

Kal inel sis ^vo evdsCag tag BPf EX ev&Elai 
nCnzovaa ^ AA tas ivalXa^ yavlag Tog vnh EAA 
A^r [0as «AAijAatff nEnoivjxtv, nagdmijlos apa ^tfiW 
17 EAZ Tfi Br. 

jJia rou do&ivTog «p« erjfiEiov rotJ A Tfj ffoftttffjj 
EV&eC^ Tij BF xaQallrj^os ev&sta YQami^ ^xTai ij 
EAZ' onep jtfft xoi^eai. ■ 



>■?'■ i^ 

Ilavzog TQiydvov [iias Tcav ni,EVQeov rtpoff- 

16 tx^^Tj&Eiarjs rj ixTog yavia Sval rats ivTog 

xaX «nsvavxiov ierj larCv, xal at ivTog %ov 

TQiyoivov zpstg yoivCai dvelv op&aig fuat sleiv. 

"EoTa Tffiymvov rd ABP, xul nQoeEX^E^k^tia&ro 

avTov fiia nlevi/a ^ BP inl ro ^- Xiya, ort 17 ixtbs 

20 yavia 7) vnu ^V^ fatj iatl Svel tafe ivTos xal a%- 

ivvn>xiov xats vno PAB, ABV, xal at ivTos rov tqi- 

ymvuv XQEtg ytaviai at iijto ABF, BPA, PAB 

(jpfratj feat Eieiv, 

'liX&m j'(rp dia toxi F Oriiitiov tjj AB 
^fi nafftt}i,Xijkos ri FE. 



XXXII. klnx. Aphrod, iu top. 
Pbilop. in iniJ. 11 p. 6&. pBellu 



t,/ AB si»BM 

)lic inph;a.foL^H 
MtiDB p. 363, 6. "H 



iiiTnl o^ii» F. T(ol soprii m. - -- _,,,..- 



uulgo. b. EA\ i: 

Bbp. T. ini] niK, 



, roo. Fi aupift m. 8 F. 



bI Bi ^fls 
'.Uijias b. 



ELEMENTOBUM LIBER L 77 

sumatur ia BF quoduis punctum 2^^ et ducatur 
ji^. et ad ^A rectam et punctum in ea situm A 
angulo ^^Faequalis construatur ^AE [prop. XXIII]. 
et producatur EA in directum, ut fiat AZ^ et quo- 
niam recta AA ia duas rectas BF, EZ incidens an- 
gulos altemos EAA, A^F inter se aequales effecit, 
erit EAZ rectae BF parallela [prop. XXVII]. 

Ergo per datum punctum A datae rectae BF 
parallela recta linea EAZ ducta est; quod oporte- 
bat fieri. • 

xxxn. 

In quouis triangulo quolibet laterum producto an- 
gulus extrinsecus positus duobus interioribus et oppo- 
sitis aequalis est, et anguli interiores tres trianguli 
duobus rectis aequales sunt. 

Sit trianguhis^BF, et producatur quodlibet latus 

eius BF bjSl A dico, angpilum ex- 
trinsecus positum^Fz/ aequalem esse 
duobus angulis interioribus et oppo- 
sitis FABy ABFy et angulos interiores 
tres trianguli ABFy BFA, FAB duo- 
bus rectis aequales esse. 
ducatur enim per F punctum rectae AB parallela 

niico£ri%Bv] BF; nsnoCrinB uolgo. 9. EAZ] EA eras. P. 

BF] corr. ex BJ Y; BVJ F. 12. EAZ] AEZ F. 14. 

tmv nXsv(fmv] snpra m. 2 F; nlsvifcig Proclos. ^ nQOCinfiXrfisi- 
crig] n^oa- add. m. 2 V. 15. i%t6g tov tQiymvov ymvla Svo 
Proclos. 16. dnsvavtiag p. iatlv tarj Proclns. iativ] 

PF; comp. b; iati uulgo. at] m. 2 V. 17. tqBig] om. 

Proclus. dvaiv] 6vo Proclus. 20. iativ P. dvai] tatg 

9val V. ansvavtCae p. "il, TAB] AVB F. at] om. F; 
m. 2 V. 22. «n m. rec. P. BVA] supra m.2 P. 24. 

sv^Bitf] mg. m. 2 V. 




78 HTOrXETQN o 



Kai ixBl irftpaAAjjAos istiv tj AB rrj FE, xal elg 
aiizuq dfm^TtTOJxtv 17 ^T, af evkXIu^ yoivCui al vitb 
BAF, j4rE fOfft «AA^Afftg iieiv. XttXtv, iml ^a^dX- 
Ai;Zos ieriv ri AB t^ FE, xal etg avras ifnt^nrmxfv 
5 si&eta 7} B^, t} ixrog ycjvia t] vitiy EFid tat} iistl 
Ttj ivtog xal anEvavTiov Tfi vxo ABF, iSeij^^&i] Sh 
xal ^ v«6 AFE Tfj usro BAF fffTj ■ oXrj aqa ^ vao 
AFJ ymvia fffij isxX 8v(i\ ratg ivTog xal aXEvavrCov 
Tatg vno BAT, JBT. 

10 Koivij^ffoexiCa&a 7; vno AFB- cct aga vao AF^, 
ATB rptei taig vno ABF, BTA, FAB tCat tiaCv, 
dXX' ai vno ATA, ATB Svfflv oQ&atg Heai tlaiv xal 
ttC imo AFB, FBA, FAB «pa Svelv opdaig laat 
tleCv. 

15 IlavTos aga TQiydvov [itag rcdv TtXevffiDV jrpoffs»- 
^Xrj&eCerig rj ixTog yeyvCa Svel tais ivrog xal dnevav: 
tCov t0ri ieriv, xal td ivrog tov TQiymvov TQsig yto-* 
viat Svalv oQ&atg t6ai elffiv oitei} SSsi. Sst^i 

Xy'. 

20 Al Tag ttSas zb xal TtaffaXXrjXovg iicl vAi 
ttvza ftipr} ijtL^svyvvoveai ei&etai xal avx' 
tcai T£ xal aaQdlXrjloi eCaiv. 

XXXin. Boetius p. 383, 11. 

3, tlaCv] PFi comp. b; elaC oi^go. ' i. iaziv] om. B. 
E r P. 5. tidEta] -v& erae. V. toj)] tai] V (>j in raa.). 
iarlv P, ut lin. 8, 8, dirivavrCas p. 7. B/l F] corr. 

rJB m. 2 V; Utt, BA in ras. B. 8. ymWa] P; Ixros yiovCn 
Theon (BFVbp), Campaima, aittvarcCits p. 10. AFE 

ABF V; corr. m. 2. U. ATS] litt. TB e corr, F, ABt 
BTA] in ras. F. FAB] om. F; BAT B et V m, 2. -~^ 

tlelv] PBF; comp, b; e/a/uulgo, 13, AFB] ABV Y (eu 



1 



ELEMENTORUM LIBER I. 79 

FE, et quoniam j4B rectae FE parallela est, et in 
eas incidit AT, anguli alterni BAF, AFE inter se 
aequales sunt [prop. XXIX]. rursus quoniam AB 
rectae FE parallela est, et in eas incidit recta B/1y 
angulus extrinsecus pbsitus EFA aequalis est angulo 
interiori et opposito ABF [prop. XXIX]. sed demon- 
stratum est^ esse etiam AFE ^ BAF. quare 

ArA = BAr+ ABr 

interioribus et oppositis [x. hfv. 2]. communis adiici- 
atur AFB. itaque 

Ar^ + AFB — ABr + BFA + FAB [x. Iw. 2]. 
uerum AFA + AFB duobus rectis aequa^es sunt 
[prop. Xm]. itaque etiam ATB + FBA + FAB 
duobus rectis aequales sunt [x. iw. 1]. 

Ergo in quouis triangulo quolibet laterum producto 
angulus extrinsecus positus duobus interioribus et op- 
positis aequalis est^ et anguli interiores tres trianguli 
duobus rectis aequales sunt; quod erat demonstran- 
dum. 

xxxm. 

Rectae rectas aequales et parallelas ad easdem 
partes ^) coniungentes et ipsae aequales et parallelae 
sunt. 



1) Hoc est: ne coniungantnr B et F^ J ei A] u. Proclus 
p. 386, 15. 

b, V (eras.), p. FBA] ATB F; BTA V (eras), Pbp. 

uQo] mg. m. 2 V. stciv tcai p. 14. bIgCv'] rFV; comp. 

b; bIcC uulgo. 17. kaxlv] PF; comp. b; hzi uulgo. ym- 

vCat, XQ^iq F. 18. dvcCv'] ycavCai <p. 20. nciQaXXiiXovs sv- 

d^eCag Proolus. 21. xal crvTa^] mg. m. 2 V. 



80 ETOIXE12N a'. ■ 

"Eataxsav tOat te xal JiapreAijjiot at AB, VJ, xal 
ijti^vyvvTiJoaav avtag ial t« avta fiEpi] sv&etai at 
AT, BjJ- liye), oti xal at AF, Sz/ leai tb xaX na^- 
alX^i.oi siOiv. 
6 'EjtE^tvx&co ^ Br. xal dxfl jrapaAiijAog ieriv rj AB 
tij r«^, xal sis avta^ efixi^tiaxtv •>] BV, at svai-Xa^ 
yatviai at vjto ABr^BF^d fflat aXk^^Xaig bIsCv. xal hxsl 
ia-ri hetlv 7] AB t^ FA xoii^ dh i] BT, Svo dij alAB^BF 
Svo tttte BF, FA^ l'aai bIgIv xal yavia ^ vab ABF 

10 Ycavia t^ vtio BFA tat]' ^aSig a^a ^ AF ^aati tfj 
BA kettv fffjj, xal to ABF Tpiymvov tra BFA tqi- 
ymv^ {eov lativ, xal at /omaJ yavCai xatq komals 
yaviaig Caai leovTOi ixatdffa ixatE^a, vip' ag at Ccai 
xXevQul vzotEivoveiv' Earj aga tj vao AFB yatvia tfj 

15 vno FBid. xttl kml elg Svo Ev&siag tas AF, B^ 
BV^cta EfiTciittovaa 15 BF rag ivaXla^ ymviag tOaq 
aXX^Kaiq ^inoitiXEv, irapttAAijAog «pa katlv tj AF ty 
B^. ISEix^ii Se avt^ xal tatj. 

At «pa tag taag te xal jcaQaXX-^lovg exI ra t 

20 fiEQ''] EXL^Evyvvovaat Ev&Btat xal avtat taat tB : 
TtaQttXlriXoi tiaiv ojiEp idtt SEiiat. 

Xd'. 
Tav naQaXXtjXoypdfifiav %(OQiiav at ditfv 



XXXIV. BoetJna p. 383, 13. cfr, PBeilnB j 



1. r^]iiiras.V. «bI— 2. eofrsf-] in, ras. b. 3, B J] (prinB) 
in raa. V. AF] PA BF, V m. 3. tt] om. FV, in ras. m. 1 
P. 6. ^J yoo ^ V m. 2. 6. rj\ in raa. b. 7. rfoAl 

PF; comp.b; efo^uulgo. 8. foi;] jj eras. V. 9. SyM- 

FBp. slttiv} PF; comp. b; flei aulgo. 10. finj i<it£ V^ 

11. i<rriv foijl fojj ieti V; ffljj p. BTJ] B JT p. 

ini*] PFV; comp, b; om, p; iazi B. 14, AFB] ABT 



i 




ELEMENTORUM UBER L 81 

Sint aequales et parallelae j4B, 
r^y et coniongant eas a^ easdem 



partes rectae AF, BJ, dico^ et- 
iam AF^ B/1 aequales et paral- 
lelas esse. 

ducatur BF. et quoniam AB rectae Fd parallela 
esty et in eas incidit BF^ anguli alterni ABFy BFd in- 
ter se aequales sunt [prop. XXIX]. et quoniam AB^Fdy 
communis autem BFj duae rectae ABy BFduabus BFy 
r/l aequales sunt, ei LABF^ BFJ. basis igitur AF 
basi B^ aequalis^ et triangulus ABF triangulo BFA 
aequalis est, et reliqui anguli reliquis angulis aequales 
erunt alter alteri^ sub quibus aequalia latera subten- 
dunt. itaque L AFB -« FBA [prop.IV]. et quoniam 
in duas rectas AF, B^ incidens recta BF angulos 
alternos inter se aequales effecit^ erit AF rectae B^ 
parallela [prop. XXVll]. sed demonstratum est^ ean- 
dem aequalem ei esse. 

Ergo rectae rectas aequales et parallelas ad eas- 
dem partes coniungentes et ipsae aequales et paral- 
lelae sunt; quod erat demonstrandum. 

XXXIV. 

Spatiorum parallelogrammorum^) latera angulique 



1) H. e. rectis j>arallelis comprehensorum. nomen ab ipso 
Euclide ad similitudinem uocabuli ev^&vy^afifioff fictum est; u. 
Proclus p. 392, 20. Studien p. 35. 

in BTA m. rec. b. 15. Post FBJ in p add. -h dl vno BAT 
TB vno Bjr. AF] AB in ras. F. 16. ycavias] P; yatvias 
xas vno ATB, FBJ Theon? (BVbp); in F rag vno ATB, 
FBJ in mg. sunt, sed m. 1; habet Campanus. 17. nsnoiri%B 
Vb. iativ aga (compp.) b. 18. Sb] 8h xal V. xaij 

m. 2 V. 

Enolides, edd. Heiberg et Menge. 6 



rroixEiaN a 



T('oi' jclEvgaC te xal yavCrtv fffat aXi.^i.a, 
eieiv, xal i) didiiEtgos avtK di^a refivtt.. 

"EOxe) 'XKQalkiil6yQu(i^fiv ifa^lov to AF^B, Sid- 

[ittffog dh uvxov i] BF' Itya, oti tov AT^B nap- 

a?.Xrikoyqd(i.(Lov ui amvavtCov nXavgaC rt xal ytavCai 

teai ai.kriXaig tieiv, xal tj BV dia^tetQog avzo Sixti 

ti(ivti. 

'EntX yag aaQaXktjlos iattv »} AB rii T/}, xot 
ti^ a\ftag {(latTttiaxtv Bv&ela tj BT, aC tvakXa^ ^co- 

10 vCttt aC vxi) ABF, Br.J teat dXXijXats bCoCv. adltv 
fitai aaQaXX^Xog eettv ^ AT tij B^, »al tig avtag 
iftitdnzaiXBV ij BF, aC ivaXka^ yavCat ai ujro AFB, 
FBj^ taai dXX^^Xtcts tisiv. Svo Sij TgCyava iati tit 
ABF, BF^ tag Svo ymvCas tag imo ABT, BFA 

15 Svel taig vnb BFA, FB^ toag B%ovTa Exati^ 

tdffa xal (tCav TCisvQav ftia nXcvQa CGtiv tijv xqqs 
tats Csaig ytovCaig xotvijv avttov tijv BF' xal tas 
Jlotna^- aQa jiAEvpaj talg Xoiaatg fffag e^et ixattQav 
ixat^pa xttl tifv Xotnijv yaviav r;} Xoix^ yavia' faij 

20 Kpa 5j (levAB nXtvQa tij FA, r} Si AF ffj Bz/, xol 
iti fetj ^ezlv ij iiito BAF ycovCa ttj vnb V.JB. xal 
ineX (0jj ietlv ij yi^lv vjcb ABV ycavCa rjj ijroo BVjJ, 
^ Si vTcb VBi^ tjj vzb AVB, olij aQa ij vab AB^ 
oXti tj} vnb ATA iauv Carj. iStCx?ti\ 8i xal rj vab 

25 BAT ty iinb TAB Cet]. 



} 



i 
1 



. dXl^Xoie b; corr. m. reoens. 9. rlaiv] PBF; comp. b; 
Lulgo. avia] -d in rnB. F. 3. AFJB] FJB litt in 
)1 litt. -rfBcorr. ei BJ m.2V; AHrj P; itemPVlin. 4. 
iij oni. p. 6. d).X^loiq b; corr. m. rec^ _ elaivj PFj 
lulgo. iixa avio p. 9. avxde) 



BumptH ob pergam. mptum in F. 1(1. tW*l PP; 
nulgo. 11. BJj JB F; B.J post raB. 1 litt (, 



(r?) \ 



A- J 
M ■ 

i 



ELEMENTORUM LIBEH 1. 



m 



II 



opposita inter se aequalia sunt, et diumetruB ea in 
duas partes aequales diuidit. 

Sit apatium paraUelograiuuiuu] Ar^JB, diametrus 
autem eiua BT. dico, parallelo- 
grammi Ar^S latera augulosqne 
opposita inter se aequalia nme, et 
diametrum BF iii duas partes 
fteqnales id diuidere. 

□am quoniam AB rectae FA parallela est, et iu eas 
incidit recta BT, anguli altemi ABF, BF^ iuter se 
aeqaales sunt fprop. XXIX]. mrsua quoniam AT 
rectae B^ parallela est, et iu eaa iucidit Br", alterui 
anguli.^r'B,ri(^ iuter se aequules auut [prop. XXIXj. 
itaque duo triauguli sunt ABT, BFA duos angidos 
ABr, BTA duobus BT^, VBA aequales habeutfs 
alteram alteri et uuum latus uui aequale, quod ad 
angnloa aequales poaitiun eat BF eorum commum^i. 
itaque etiam reliqua latera reliquis aequalin habebuut 
altemm alteri et reliquum angulum reliquo angulo 
[prop.SXVI]. quare AB-^TA, Ar = BJ, LBAT 
'-=r^B. et qaoaiam LABT^ BPA ei FBJ = ATB, 
erit i AB^ = AV/i \x. ivv. 2]. aed demonstratum 
eet, eBse etiam £ BAT <^ r^B. ergo .spatiorum par- 
allelogrammorum latera augulique opposita inter ae 
aequalia sunt 



ArS\ BFJ F. la. tW»] PF; comp, b; fia 

PF; comp. b. Ttr] tiSF. U. ETJ] in rog. m.aV; i'iiJ 

F_ 16. rj nti Y. 18. Xomaig Ttliofats FV. 21. Iti 

i^H] tWv] P; om. Theon (BFVbp), r*JB] BFJ p. —' 

imti— Sa, Sr^] mg. m, recenti p. <><.■-'■> -i i-'" 

corr. V m. 2. APB] litt. TB e ce 

— »0. r<nt] rag. m, « V. 



1 
I 




33. rs^] iiU rs 

V m, a, 24. ISiix^ 



84 I^TOIXEIGN a. ■ 

Tmv aQa uaqaXXiiKoyQa^L^iQiv xisqloiv at uiiavttv- 
tCov iti.EVQttC XE xcd yojvCtti £3iu aXl^laig ElaCv. 

Aiya S-fi, ort xaX 17 Sia^EiQQg avta SC%a tE^vet. 

s«el yaQ t<S7} EStlv ii AB tfj F^, xotvii Sl i\ JSF, 

'■• 6vo Si} aC AB, BF SvgI ttt^s FJ, BT tOai. «Otv 

ixazEQtt ixatBQ^' xal yavCa ij vno ABT ymvCa tfj 

vTco BF^ faij, xttl jSKffig apa ij AF t^ z/B fffjj. xal 

T(i JBF [ttQtt] rpCytovov ra BF-J tQtydv^ ttsov iOtCv. 

'H aQa BV Sid^BTQog SCxa ti^vtt th ABF^ 

10 nttQai.hjX6yQttfi(iov oaeQ iStt dEi^at, ^J 

W. t 

Ta ntaQaHjjloyQaitiia ta ial t^j avr^g ^d- 
aeas ovta xal iv ratg avtaig iiaQttlX^Xots COa 
ttXXifXois iotCv. 

i& "Eerm nttQalXr}X6yQafintt ra ABV^, EBTZ inl 
T^s avtris ^deEos tfig BF xal iv ttttg avzats nttQttX- 
J.)}Aocg rats AZ^BF- Xiya, ori [0ov iotl zo ABTjJ 
ta ESrZ jtaQaXX7)XoyQdfi(i^. 

'EtieI yuQ aKQ«XXriX6yQaii(t6v iatt ro ABFjJ, fff»j 

no i<stlv ii AA rfi BF. Sta ra avta Sij xal ^ EZ ty 
Sr iattv fffj;' met£ xal ^ A^ zfj EZ iartv [ej}- xal 
xoivi ^ JE- oA?) ttQtt 7} AE 0% rfj AZ iffttv iffTj. 
lati, Si xttl ii AB rfi ^r rerf dvo Sij ttl EA, AB 
dvo ratg Zdi, ^F teai dalv ixtttEQa ixariQ^' xal 

25 yetvCa ij into ZJV yavta tfj vno EAB iattv tarj q 



XXXV. Pdellus ]>. 46. Boetius p. 383, 17. 



2. ctaC B. 3. Si'] om. P; oorr. ex *e m. 3 V. 6. rJ\ 
BT] BF, in raB. m.sVi ^jr, rs P (jrin ras.); Br.FJbp. 
7. ya/l om. p, oeal om. P. xr}\ paa*. xij p. JB] BJ 
P etV, sed corr. m.2. tar,] P; ioiiv fffij Theoii(BFVbp). 



ELEMENTORUM LIBER I. 85 

iam dico^ diametrum ea in duas partes aequa- 
les diuidere. nam quoniam AB — F^ ei BF com- 
munis; duae rectae AB, BF duabus FJ, BF aequales 
sunt altera alteri; et L^Br^Brj [prop. XXIX]. 
itaque etiam {Ar^JB, et]^) A ABF = BTJ 
[prop. IV]. 

Ergo diametrus BF parallelogrammum ABF^ in 
duas partes aequales diuidit; quod erat demonstrandum. 

XXXV. 

Parallelogramma in eadem basi posita et in iis- 
dem parallelis inter se aequalia sunt. 

^ JS j g Sint ABTJ, EBFZ paral- 

lelogramma in eadem basi BF 
et in iisdem parallelis y^Z^^r*. 

jf T <iico, esse ABFJ — EBTZ. 

nam quoniam parallelogrammum est ABFjdy erit 
A^ = Br [prop. XXXIV]. eadem de causa etiam 
EZ = Br [id.]. quare AA = EZ [x. ivv, 1]. et com- 
munis est AE, itaque AE = AZ [x. ivv. 2]. uerum 
etiam AB = jdF [prop. XXXIV]. itaque duae rectae 
EAy AB duabus ZA, JF aequales sunt altera alteri; 
et L ZAr= EAB exterior interiori [prop. XXIX]. 

1) Fortasse potius xttl f^tSicig apa rj AF t^ JB tari lin. 7 
delenda sont quam apa lin. 8 cnm Angasto. 

8. Sga] del, Angast. BTJ] BjrF^BATh.BedA eras. 

toov iat£v'] PBb (comp.); taov ^atai FV; imv faov p. 
10. Post naQalXrjXoyQafifiov in V add. %ioq£ov, sed pnnctis del. 
m. 2. 13, ovxa'] om. Proclus solus. 17. iaxiv P» ut lin. 

19, 28. 18. naQalXriXoyqdiiiiqi] P; om. Theon (BFVbp). 

20. *ij] mg. yg, xoivvv F. h\j^' 2 F. 22. htiv'] om. F. 
28. EA] AE F. 24. ^vai BVp. ZJ] JZ¥, 26. ^] 
(alt) supra m. 1 P. 




86 STOIXEinN c 



1 



ixzog T^ ivzog' ^aSts apa 7] EB ^«.asi t^ Zf ('tfij 
iarCv, xal ro EAB ziffytovov ta ^ZF rpiymvip i'6ov 
sOtaf xoivov aqDfip^ff^iB to jJHE' i.oixov apa ro 
ABH^ TQcmi^ov Aociito zip EHFZ zQKXE^iia iazlv | 
5 i6ov xoivov jtQOSxsiaQ-a lo HBF t^iycavov oAcw 
«p« To ABF^ nK^tti.irjl6yifaniiov oAp wo EBTZ 
xaQaXJLrii,oypa(i[ia iSov BOziv. 

Ta UQtt iiagttXXr}k6yQKii(ia za ini r^g rut^s 
0£OJS ovTB x«i iV rais «iJTars JtapaAAjj/oig fOa «AA^ 
10 Aots ioriv' oTCtp Idtt dst^ai. 

ke'. 

Ttt «a(fKlX7]X6yQa(iiia za ixl Satav fiaOsa»' 
ovta Kal iv Tatg avTatg itaQalX^lotg tOa ai.-^ 
AfjAois iSTiv. 

l& "EtfTm jrapaAAijAoj^pK^fta t« ABV^, BZH0 ^«li 
riJrav ^ttSiiav ovtk Tmv BT, ZH xal iv rarg «vzats> 
xuQaXlilloig rulg A®, BH' kiyo), ort fffov iffTi ti' 
ABTjd 7[aQakXt]l6yQa(iftov tcj EZH@. 

'Eaetevx^atCav yag al BE, T©. ;«i! ^fi fffl(' 

20 ioilv 7] BF r^ ZH, dUa ij ZH zy E@ ioztv fffij, 
lud ■^ Sr aQK T^ E& iOTiv lori, tial Sl xal xaQ«ir 
Ai]Ao(. xai im^evyvvovaiv avtag ccl EB, ®r' aC Si 
tas i'6ag te xkI naQttXli^lovg ijcl za avTU [ligii i^t- 
^evyvvovaai LOut tb xm. xaQaXi.r}XDi siat [xai al EBfi 

H5 &r ttpa raai ti aioc xal actQaXXtjXoi]. jrapaAAjjAiJ-' 

XXXTI. BoctiuH p, 383, 19. 



1. Zr] mutfit. in rz ni. 2 V, 2. htiv] PF (in B y ec&s^ 
uomp.b; ifltt' uulgo; letiv fit^ p. JZrl BF, Y m.aj JTSfc 
P; Zjrbi., Vm.l. 3. farat] PBFpj Jcrri^Vb, xo] port- 
ea add. P, JHE] corr. ej JH P; iwo JHB F; ««* 



ELEMENTOBUM LIBER I. 87 

itaque EB — ZF et A EAB -= JZF [prop. IV]. 
tnbtrahatary qni communis est, triangulus /IHE. ita- 
que ABH^ «» BHFZ [x. Iw. 3]. oommunis adiicia- 
tnr triangulus HSr. itaque ABTJ^^EBrZ. 

Eigo parallelogramma in eadem basi posita et in 
iisdem parallelis inter se aequalia sunt; quod erat de- 
monstrandum. 

XXXVL 

Parallelogramma in aequalibus basibus posita et 
in iisdem parallelis inter se aequalia sunt. 

Sint parallelogramma 
ABTJ, EZHe in ae- 
qualibus basibus BF, 
Z ff et in iisdem par- 
J^ r Z M allelis A», BH. dico, 

esse ABFA =« EZHe, 

ducantur enim BE, r@. et quoniam BF^ZH 
et ZH = E0y erit etiam BF = E@ [x. ivv. 1]. uerum 
etiam parallelae sunt. et coniungunt eas EB, SF] 
quae autem rectas aequales et parallelas ad easdem 
partes coniungunt; aequales et parallelae sunt [prop. 
XXXni]. itaque parallelogrammum est EBFS [prop. 

eras. Vb. inaomov P. 4. EZFH F. 6. jBT JBTJ BHF 
F. 7. iativ] PF; comp. b; iar^ uulgo; om. p. 8. &Qce] 

&UM Y; corr. m. 1. 18. iatlv dlXriXois p. 14. iat£ Pro- 

clus. 17. BH] HB F. iativ PF; comp. b. 18. EZH9] 
Pb, V (E e corr.); ZH9E BFp; in V sequitur ras. 1 litt. 
19. BE] EB P. r&] in ras. P. 20. BT] Pb, V e corr. 
m.2; rs BFp, V m. 1. aXX' F. dXXa ^J mg. m.2 V. 
21. siatv P. 22. BE, r® b, V e corr. m.2. 23. Tel om. 
P. 24. ti elat %al nocQtxXXriXoi F. xat J (ali) om. F. 

xai a£ — 26. nagdXXriXoi,] xal ai £B, QF a^a laat xb xal naq- 
dXXfjXoi slai P. m. rec. 24. EB] E insert. m. 1 V. 25. 

er] V m. 1; re v m.2. 




88 ETOIXEIflN a. 

yfittiiftov ttQtt ietl 10 EBr&. xai ietiv ^eov ta ARF^' 
^tt<Si.v TS yccQ avra rijv kvtiiv Sibi tiji/ BF. xal iv 
Ttttg avTats aaQaHLtjkotg iezlv avta Tatg BP, A@, 
dia Ttt ttVTtt Srj xal to EZH® tS aita rra EBr® 

5 i>STiv fffow &6zi xaX To JBr^d JCaQtti.).T]2.6yQanfLov 
zm EZH& iCTiv Caov. 

Ttt apa attga^XTil6yffttii{itt tu inl Ceoiv ^<i6Eav 
ovTK xttl iv Tttig ttvratg TzaQaXk^Xoig tsa a},i.riXois 
iarCv okbq idsi Sft^at. 

Ttt tQLymva r« iTtl r^g ttvtijg ^aaimg ovra 
xal iv Xtttg avTtttg jrKpciijjAoig ioa KAA^Aotff 
iativ. 

"Eetsa rQCysavtt ta ABF, ^BF izi r^g avrijs ^d- 

6 6ct3g T^g BV xal iv talg avtatg 3taQtt'kXT\'Kotg tatg 
A/i.,Br' Xiya, ort foov iatl ro ABF TQiyatvov rtS 
.JSr rgiyavm. 

'ExpepX^a^m ij AA isp ixdzeQa ra ficpij i^l ta 
E, Z, xal Sid filv zov B r^ FA naQaXXijXog rjxQot 
^ BE, dia Si rou F zrj BJ naQaXXrjXos %*ei rj FZ. 
nttgaXXijXoypaiiiiov aga istlv ixaTSQov rmv EBFA, 
ABFZ' xttC tiaiv laa' ini re yaQ Tijg ttvtijg ^aaccog 
eiat Tijg BF xal iv Tatg avTats JtapttAAjjAotg rffFs 
BF, EZ' xaC iari tou filv EBFA ■scaQttXXijXoyQttfi- 
25 p,ov rjfiiav ro ABT tQCyavov ij yaQ AB SidfisTQoe 
ttvTo Si%tt zifivei' Tow Si ^BFZ utaQttXXtiXoyQdfifiov- 

XXXVII. BoetiuB p. S83, 22. Apud Proolum eicidit, 

1. ItnCv PF; comp. b. _ Tri] eorr. es lo m. 1 V. 
Ivtiv Vttlialli]lois p. i. av-cm tco] mg. m. I Fi om. p. 



^ 



I 
1 



ELEMENTORUM LI6ER I. 89 

XXXIV]. et ESre = ABTJ] nam et eandem ba- 
sim habent £F et in iisdem parallelis sunt BF, AS 
[prop.XXXy]. eadem de causa etiam EZHS «> EBFS 
[id.]. quare etiam ABFJ =« EZH0 [x. ivv. 1]. 

Ergo parallelogramma in aequalibus basibus po- 
sita et in iisdem parallelis inter se aequalia sunt; 
qaod erat demonstrandum. 

XXXVIL 

Trianguli in eadem basi positi et in iisdem paral- 
lelis inter se aequales sunt. 

Sint trianguli ABF, JBF 
^\ ySc '~7'^ in eadem basi^F et in iisdem 

\/^_\./^ parallelis AAy BR dieo, esse 

B r A ABT — /IBT. 

producatnr A/i va utramque partem ad £, Z, et 
per JS rectae FA parallela ducatur BEy per T autem 
rectae B^ parallela ducatur FZ [prop. XXXI]. itaque 
EBTAy ^BTZ parallelogramma sunt; et sunt aequa- 
lia. nam et in eadem basi sunt^F et in iisdem par- 
allelis BT^ EZ [prop. XXXV]. et dimidia pars par- 
allelogrammi EBTA est triangulus ABT'^ nam dia- 
metrus AB id in duas partee aequales diuidit [prop. 
XXXIV]. parallelogrammi autem ABTZ dimidia pars 

8. alXriXoiq] -Xoig corr. m. 1 V. 9. htiv] doiv F. 16. kaxiv 
P et eraso » y. In F hic uerba nonnulla enan. 19. E, Z] 
Z, E F. xal 6id — 20. BE] mg. m. rec. p. 19. TA] A 

in ras. b. 21. x&v] v postea add. m. 1 V. 22. /^BrZ] 

BdrZ F. fiaiv toa] P; faov ro EBTA xm JBTZ Theon 

(BFVbp; BdFZ F; in EBVA Htt. EB m. 2 V). t«] om. 
Bp (in F non liquet). 23. doi] Bbp; dGiv P; hxiY\ iaxiv 
F. xais] (alt.) iaxlv xaig F. 24. BF, EZ %aij absumpta 
ob ruptum pergam. F. ioxiv P. 25. xo] xa in ras. P. 
26. naQaHfjXoYQaniiov] mg. m. 2 V. 



90 STOlXEIiiS a'. 

ijfiiSv To ^BV zQtyavov 7} j-ap ^F diafitt^og avro 
Sixct riiivst. [ra Ss zmv iSav ij[i£aij ("(Ja aXlijloig 
iatCv}. loov apa ierl ib ABF tffiyoyvov rp ^BP 
XQiymvG}. 
5 T« aga TQtyavtt tcc i^l tijg a^T^g ^affsmg ovra 

xal iv taig avtais TtapaJ.K^^lotg lOa a^AiJZotg iattv 
onsQ idei Ssi^tti. 

Ta Tffiyava tic iitl tOav ^aSaav ovxa xal 

10 iv tatg avtatg naQalX^^Xois tSa «iiijAotg istCv. 

"Eetm r^Cyaivtt ta ABF, AEZ i%l iaatv ^aStcav 

tav BT, EZ xal iv talg avtals JtapaAAiJ/otg rtttg 

BZ, AA- Xiya, ott fffov istl ro ABP ZQiyavov ta 

AEZ TQiyinva. 

15 'E«/3£^/iJofrra ya^ ^ AA iip' ixdzBQa ta [tsQii i%l 

ta H, ®, xal Sta (ilv tov B t^ VA naQaKltjXos ijx^oi 

71 BH, Sta S\ row Z rri d E TcaQaXXi^Xos ijx&cj 17 

Z®. JtuQaXXi^iXoyQafiiiov uQa iatlv ixdrsQOV tcav 

HBFA^ AEZ&- xtti taov to HBFA tm ^EZ®' i%C 

20 ts yag iOav ^duiwv «fft tav BF, EZ xal iv ratg 

avtatg jtaQaXXi^Xois T:atg BZ, H®' xaC isrt tow (*W 

HBFA 7tttQttXXijXoy0ttii[iov yfiiev ro ABF tQCycavov. 

ij yaQ AB Std(tstQos avro SC%a tsfivsf roC Ss ^EZS 

naQalXf]XoyQdfifiov ^(itSv tb ZE^ tQCyavov i] yttff 



SXXVIir, Boetiua p, 383,2i. 



< 



1. JSr] dTB F. tptycavov) aupra m. 2 V, Jrj 

abBumptam in F. 2. oli^Ioie] Bupra m. 2 V. 3. loi^» P. 
9. lamv] PBV, Proclns; rm»' foojv Fbp; cfv. p. 86,12. i'oini' 
in raB. p. 10. hriv] PVp, Proclns; eUi> BFb. 11. JEZl 
corr. ejt Z^E F, Pooio..] PBp; ^.iaiav 6tra Fb, V (sei 

oi-to puBBtiB del. m. 2). 12. EZ] oorr, ei ZE F, 13, 

;«('»> P. 15, Ul] xam P. 16, 1^] corr. ei x^e V. 



ELEBfENTORUM LIBEB I. 91 

«st tarUuigaliifi ^BF] nam diametrns ^F id in duas 
partes aequles dimdit. itaqne^) A ABr= ^BR 

Ergo triangnli in eadem basi positi et in iisdem par- 
allcAiB inober se aeqnales snnt; qaod erat demonstrandum. 

xxxvin. 

Trianguli in aequalibus basibus positi et in iisdem 
parallelis inter se aequales sunt. 

M ^ ^ 6> ^"^* trianguli ABF^ AEZ 

/ ^/7 rv \ in aequalibus basibus BF^ EZ 
//^ / \ NJ et in iisdem parallelis BZjAd. 

» T Jf Z dico, ease AABr^^AEZ. 

producator enim ^^ ad utramque partem ad H^ 
S, et per JS rectae FA parallela ducatur BHj per Z 
autem rectae ^E parallela ducatnr ZS [prop.XXXI]. 
parallelogramma igitur sunt HBFA^ JEZ0. et 
HBFA « AEZS\ nam et in aequalibus basibus sunt 
BF, EZ et in iisdem paralleHs5Z, He fprop. XXXVI]. 
et parallelogrammi HBFA dimidia pars est triangulus 
ABF^ nam diametrus ^^ id in duas partes aequales 
diuidit [prop. XXXIV], parallelogrammi autem AEZ0 
dimidia pars est triangulus ZEA\ nam diametrus AZ 

1) Gain constet, x. ^w, 6 ab Euclide non profectam esse 
(cfr. f^oohis p. 196, 25), quamquam tempore aatia antiquo (ante 
Theonem saltem) interpolata est, ueri simile est, uerba xa di 
%mv tcmv TjfUari i'aa dllrjlois icxCv lin. 2 et p. 92, 1 eodem tem- 
pore irrepsisse. Eudides usus erat x. ivv, 3. 

17. HB P. Z] E P. zfE] Ed P. 18. ZO] E« F. ~ 
19. dEZB] (prius) JVES P. 20. xc] om. p. x^v tvmv 
p. bIciv PB. t(dv] oorr. ez tcm m. ^Y. EZ] ZE e 

corr. P. 21. BZ, HB\ BH, ZS V; corr. m.2. icxiv P. 
28. xov di — p. 92, 1: xipuvti] mg. m. 2 V ad hunc locnm re- 
lata. z/EZO] jrEB, E in Z corr. F. 24. ZE^] EJT 
F; /JEZ b. 



92 STOrSEI.QN a'. 

^Z diaftfZQOs uvro 6Cxet iBftvsi [tk di xmv [oav 
Tifuffr} lea a},Xrii,ots dtfziv]. leov aga iexl ro A&V 
Tffiycavov ra z/EZ rptyoivoj. 

Toc aQu Tgiyava ta ^Ttl teatv fiaofcav ovta xai iv 

5 Tcfg avrais napaXX^^Xoig lea kAAiJAois ierCv Ssrsp 

HSti dti^ai. 

Ttt i'ea zqCyava ra inl Tijg aur^g fiaeeag 
ovTa xal ial ta avta fiiQri xal iv raig av 
10 KopaiAjjAots ierCv. 

"ESTm tea rpCyava ra ABF, ^BF inl rrig avr^s 
fideEag ovra xal ial ra avta fiiQri tt^s BF- Xiyio, ort 
xttl iv Tafg avraig xaQall^^loie iarCv. 

'ETteltvx^oi yaf) ii J^}- liyca, oxi mapaAAj/Aog ieriv 
16 t/ AA r^ BT. 

Ei yap jij}, ^'x^M ^'K tov A etjjieCov r^ BP t^- 
&ECtt aaQdllrjlog tj AE, xal ixEttvx^a ^ EF. leov 
ttQtt ior} ro ABF rgCyajvov rra EBF rQiydva' iaC 
re yuQ r^g avr^g ^deemg ieriv avtm r^g BP xkJ. 
20 iv tais avrais aapa^l-^loig. dXXa ro ABF rra 
^BF ieriv teov xal ro .JBr &Qa ra EBF i'eav 
ierl to (leitov rp i).desovi' ontsp ieriv ddm'aTOv' 
ovx aQtt staQalX7]X6s ieriv 17 AE tij BF. o[ioCcog S^ 



XXXIX. Boetius p. 384,1, 



1. JZ] Pb, P e corr.; ZJ BVp. teav yo,vt6v F. 
ioTir] PVp; tleCv BPb. Imn idrlv PF; comp. k 

JEZJ corr. ei Z^E F. 6. W»] flaiv BFb- 8. tbI 

(alt.) om. b. 9. x«i i-al re aixa p,hr}\ P, F (del. m. 1), T j 

m. 2, Boatins, Proclas, CaJiipaiins; om. Bb, V m, I, p, *at^ J 
(ftlt.) oro, Ptoclus, 11. yp. 9vo mg. V. IB, o»io] om. p, 
*a\ ijt\ ta ai'ire <iiiiri\ P, Cnmpanus; om, Tlieon (BPVbp), 



B^ 



ELEMENTORUM LIBER L 93 

id in duas partes aequales diaidit [id.]. itaque 

A ^5r = JEZ. 
Ergo trianguli in aequalibus basibus positi et in 
iisdem parallelis inter se aequales sunt; quod erat de- 
monstrandum. 

XXXIX. 

Aequales trianguli in eadem basi positi et ad eas- 
dem partes in iisdem parallelis sunt. 

Sint aequales trianguli ABFy JSr 'm eadem basi 
positi BF et ad easdem partes. dicO; eos etiam in iis- 
dem para]leli8 esse. 

dueatur enim Ad, dico, AA parallelam esse 
rectae BF. 

nam si minus, ducatur per A punctum rectacjBF 

^ parallela AE [prop.XXXI]^ et ducatur 

ET, itaque A ABr= EBTi nam 

in eadem basi sunt BF et in iisdem 

parallelis [prop. XXXVII]. uerum 

^ A ABT = ABR quare etiam 

A ABr = EBT [x. ivv. 1], 

maior minori; quod fieri non potest. itaque y^J^rectae 

Br parallela non est. similiter demonstrabimus, ne 

13. iariv] sIolv p. 16. arjfisiov] om. p. sv&^ia] om. p. 
18. aQa] d?} P. iativ P. 19. iaxtv avTw] stai p. B F] 
FB F. 20. aXXd] PB, F m. 1, V m. 1,^ b m. 1; xaig JBT, 

AE. dXXd p, V m. 2, b m. 2; in F pro dX- scripsit q>: taig, 
sed -Xd rehctxim est. Post JBF add. zQLymvov P m. rec, 

VBp; comp. Bupra scr. m. 1 F. 21. taov iazl t^ dBF tQi- 
ymvip p. iativ] euan. F. JBF] (alt.) JTB F. aQa] 
om. P; UQa tQiycavov P m. rec, p. laov iatl tm EBF xqi- 
ymvq) p.^ ^ 22. iatQ iatCv PFb iatCv] PBb; om. Vp; in 
F est: ddvvatov q), sequente vatov m. 1 (fuit sine dub. iatlv 
ddvv.). 23. ofioCoag] mg. m. 2 V. 




94 



ETOIXEI.QN 



dtiloftBVf ozi ovd' alit] zig nX^v z^g A^^ ^ AA a 
t^ BT £0zi JiaQdXktjlog. 

Ttt aga taa ZQiycova ta iTcl vijs avr^g pdeeayg 1 
oi^a xal ixl ta avta ^epij xal iv zaiS uvztttg aaQaX- 
5 ^ifiotg ietiv onBQ idei Stt^ai. 



I 



Ta (sa zqCymva xa inl ieeiv ^doemv ovza 
xal inl tic avta y-iffti xal iv zaSg avtatg xag- 
aXX^^Xoig iativ. 

n "Estm Hea ZQiyava za ABF, F^E ial Hotov fid- 
oeav tav BF, FE xal inl za avza (liff^. Xiym, otl 
xal iv rarg avtaig JCtt^aXX^Xois istiv. 

'Em^svx^^o} y«p ^ Ad' Xiym, oti xagaXXijXog iariv 
^ AJ ty BE. 

6 Ei yap (i)j, tjx^'" ^'" ^o'' -^ ^S ^^ xa^dXX-rjXog 
ri AZ, xai inalsvx&m tj ZE, i'aov aga iotl to ABF 
zgiyavov zm ZPB tptycJvcj' ijii ze yaQ tOmv ^desiav 
eiei zSv Br, FE xal iv tatg avtaig xaifalX^^Xois 
tatg BE, AZ. dlXa to ABF zqiyavov fffoj/ iOrX rp 

^FE [ipt yaJi/p] ■ xal ro ^FE apa [tQiyavov'] taov 
ietl Tw ZFE tgiymvm ro fiet^ov zm iXdeaovf oaef 
iatlv ddvvatoV ovk ap« jrap«AAijios r} AZ ty BE. 
6(ioims Si) deiioiisv, oti ovd' aXXij Tig nXijV z^s ^-d' 
71 A^ dpa t^ BE ieti TtttffdXX^Xog. 

XL, Boetins p. 384,4. 

1. ovSi FVbp. 2. ieriv P. 4. x«I Iwl tu bwib ftfetj" 
om. BFVbp. 7. taov] PBVLp, Proelue; tav taiav F, sei 

imv punotiB dcl, 8. xal inl la aviu fie^i/] P (del.), V mg; 

m. 2 (tiai m. 1), Proolne, Boetiiu, CaiDpa.nQB; ani. B, V m. 1, 
bp; in F: *al ixi 7, deia post lacuiia,m pdaiig ovm m. 1, 
punctiB del. koi'] (alt.) om. PcocIob, V. 9. iexiv] iml 



1 



■-^— 




ELEMENTORUM LIBER I. 95 

aliam quidem uUam praeter A^ parallelam esse. ita- 
que A^d rectae BF parallela est 

Ergo aequales trianguli in eadem basi positi et 
ad easdem partes etiam in iisdem parallelis sunt; quod 
erat demonstrandum. 

XL. 

Aequales trianguli in aequalibus basibus positi et 
ad easdem partes etiam in iisdem parallelis sunt. 
Sint aequales trianguli AEF^ TAE in aequalibus 

basibus ETj FE et ad easdem 
partes. dico^ eos etiam in iisdem 
parallelis esse. 

ducatur enim A^, dicO; AA 
rectae BE parallelam esse. 
nam si minus, per A rectae BE parallela ducatur 
AZ, et ducatur ZE. itaque A ABT'^ ZTE] nam 
in aequalibus basibus sunt BFj FE et in iisdem par- 
allelis BE, AZ [prop. XXXVni]. sed A ABT'^ 
^FE. quare etiam A AFE — ZFE [x. ivv. 1], maior 
minori; quod fieri non potest. itaque AZ rectae BE 
parallela non est. similiter demonstrabimus^ ne aliam 
quidem ullam praeter AA parallelam esse. itaque AA 
rectae jB^ parallela est. 

Pioclus; sialv p. 10. FJE] dTE P. 11. inl xa avta 

fiSQTi] mmctia del. P; om. Theon (BFVbp). 12. iaziv] P; 

sCa£v Theon (BF Vbp); cfr. p. 92, 13. 14. EB P. 16. ZEI 
Zr P. UQa] dri P. iativ P. 17. tqfyaivov tm ZTEJ 

om. P; xqiymvov xqiymvxo xm ZFE m. rec. .18. elaiv PF. 
19. AZ, BE p. iax/v F/ 20. JFE] litfc. J in ras. m.2 
V; JEr F. XQiymv(o] otn. P. XQtyoiVov] om. P. 21. 

iaxiv P. ZrE] ZETF. 22. iax£v] om. p. iativ ^ p; 
Post A Z lacanam V. 23. ovdi p. 24. tj] in raa. m. 1 

b. iaxtv P. naQdXXriXos iatt Vb. 



96 



ETOIXEIiiN u 



Ta oiQa tea T^fyava zec ial Haav ^aaeav ovza xol i 
inl za avxa. fiSQJi xal iv xaZs avzatg JrapaAAijAots ioziv ' 
oxEQ idti Set^at. 

6 ^Eav 3Ctt^akk'^!i.6ypaii{iov r^iytovp fiaOiv i 

^XV ^V'" "vt^v xal iv tais avTaig TcagalX^^Xo 
»j, di^Xttai.6v iati to itagaXirjioyQaiiiiov tov 1 
rpiyavov. 

naQai.lijX6yQafi(iov yaQ to ASV^ TQiycavip 

10 ESr ^daiv T£ i^ETm t^v avrijv rijv BF Kal iv talg 
avtaig nttQaki.^^loig ietci Ttttg BT^ AE' kiya, qti 
SmXdaioi' ieti xo ABF^ attQaXl7)X6yQa(i^ov tov BEP 
TQLymvov. 

'E%£%Bvx^a> yuQ ij AF. taov d^ iati to ABF tq£~9 

15 yavov T» EBV TQiymva' inl re yaQ r^g avtr[q ^-f" 
d£Q)g iaziv avtm triq BV xa\ iv Tatg avzai^ aaQairj 
^iJAotg Tttig BF, AE. alXa zh ABFd aaQaXXiiXo^ 
yQaiijiov diirXdatov itSTi xov ABV tQtymvov 7] yitff -1 
AV didfiBTQog avTO di^a tinvtf Satt zo ABVj4 ' 

20 xaQaX?.7iX6yQtt[i^ov xal tow EBV TQiycovov iarl 3t^ 
jtXdrfiov. 

Eav aqa staQaXXrjXoyQafifiov tQiyaiva) ^aoiv ze ^XBm 
tijv ttvzijv xal iv zaZs avtaZs JtttQttlX^^Xois 'ri, SixXA-^% 
tftdv iati t6 aaQttXXrjXoyQaiiftov zov ZQiycovov onSff \ 

25 ISbi SBt^ai. 

XLl. Boetins p. 384, 7. 

1. la ial — 3, fffijoii] mg. m. 1 b. nal ixl u. , 

(iipjj om, PBfVbp. 3. i<!ti xagaXl-^loisV. 7. ij] aupi» -i 
m. 1 F. ioii] ProcluB; iaxiv P; cfr. lin. 21; larac BFVbgr ' 
cfr. BoetiuB, Campanua. 9. tp] m. rec. P. 10. te] 
Tfl'v] (alt.) t^( BV, corr. m. 2. in» BT] Bnpra m. 
11. i^oim jtnefWijitne V. IS. i<!Ti.v P. BET] EBP P. 



ELEMENTORIJM LIBEB I. 97 

Ergo aequales trianguli in aequalibus basibus po- 
siti et ad easdem partes^ etiam in iisdem parallelis 
sunt; quod erat demonstrandum. 

XLI. 

8i parallelogrammum et eandem basim habet^ quam 
triangulus aliquiS; et in iisdem parallelis est, duplo 
maius est parallelogrammum triangulo. 

parallelogrammum enim ABFd eandem basim ha- 
/J E beat BFy quam triangulus EEF^ 
et in iisdem parallelis sit BT^AE. 
dico^ parallelogrammum A B FA 
"fi duplo maius esse triangulo BEF. 

ducatur enim AF. itaque A ABr= EBF] nam 
in eadem basi sunt £F et in iisdem parallelis BF, 

AE [prop. XXXVII]. sed ABFA — 2 ABr, nam 

diametrus AF id in duas partes aequales diuidit [prop. 

XXXIV]. quare etiam 

ABFA = 2 EBr.^) 

Ergo si parallelogrammum et eandem basim habet, 
quam triangulus aliquis, et in iisdem parallelis est^ 
duplo maius est parallelogrammum triangulo; quod 
erat demonstrandum. 




1) Hoc ita ex axiomatiB colligitur: 

ABT^ EBF, 'lASr ^ ^EBV [x. iw, 2]. 
2ABr^ ABrj'y ergo 2EBr^ ABFJ [x. ivv. 1]. 

14. A r] corr. ex AB m. 1 F. icxiv P. tQiyoivov'] om. V 
15. EBF] E snpra m. 2 V. 16. naQctXX^nloig] 'Oig in ras., 
seq. ras. 6 utt. V. iariv P. 20. xal tov EBF rptyoovov] 
tQiymvov tov EBr Y. EBT] corr. ex ^BT m. 1 F. iatlv 
F; comp. b. 23. jj] supra m. 1 F. 24. iati] BFb; iattv 
P; ^tfra* Vp. 

Enclidei, edd. Heiberg et Menge. 7 



98 STOISEiaN a. 

Tra do&evTi tptyfova i'Oov TtapaXXr^^oygcc 
(tov SvOttjaaa^ai iv tfj do&siSri yiaviif Bvf^v—m 

"Eerm To (liv So&ev t^iymvov to jiBF, ij di So^ 
&ttaa yatvia f u^&vypajiftog ^ ^ ■ dtl tfij ta ABF tqi^M 
yeiv^ teov jiagalXviloyffajifiov evatijaae&at iv tij ^M 
ycovCa £i&vyQd[t[ta. 

TfTfwJfl^to ^ Br Slfja xata tb E, xal ^9[cg£vj;#a> 
j, AE, xal ewEtStdta ffpog ty ET sv&tia xal t^ 
jipog avtTi arjfitio} ta E rfj A ytavia iav) ^ vnb FEZ, 
xal dta fiEV tov A t^ ET TtapdXXiiXos ^S^a) 17 AH, 
dta Si TOw r tfj EZ nagdXXijXog ^x^^ V ^H' ^ctg- 
aXX7]k6ygttfi(iov apa iatl to ZEm. xal cittl tGt] 

15 iatlv ii BE tfi EF, taov ierl xal ro ABE rpi^yovov 
rai AEF tQtydvp' ini te ya^ tsmv ^daeav eiei. tmv 
BE, EF xal iv tatg avtalg JtagaXXi^Xots tatg BF, 
AH- diaXdatov opa iatl to ABrtQiymvov totl AET 
tQiyiavov. iSti 8\ xal to ZEFH xagaXXriXoyQafiftov 

20 SiJtXdatov tov AEV tpiycovov ^datv tt yag avtp 1 
■tijv avtijv i%tt xal iv tat^ avtatg iativ avt0 TtagaX- 1 
X-^Xoig' taov aga ietl to ZEFH TtaQakXfjXoyQaftnov i 
Tp ABF tQtymva. xal ix^i tijv vmo FEZ ymviav 

Cetjv T^ So&tia^ji tfi A. 

Tp ttpa So&ivzt TQtytovm tp ABF teov aaQaX- 



XLII. UoetiuB p. SB4, 13. Apad Pioclam excidit in codd.; 
floetitie prop. XLIl— XLIII permTitanit. 

3. avar^^aiia&ai] otiocijanct ip (F ODOiiJflflff&ei). iv] Iw i 

■/amCif, ij ioTiv fet] ei Proclo in prop. XLIV recepit Aaguat I 
Bnadente Qregorioi ofr. CampanaB. 7. 1^] P "^- ^i ^^< ^M 




ELEMENTORUM LIBER I. 99 

XLH. 
Dato triangulo aequale Iparallelogranimum constru- 
ere in dato angulo rectilin^o. 

Sit datus triangulus ABF, datus autem angulus 
rectilineus ^. oportet igitur triangulo jdBF aequale 
parallelogrammum in angulo rectilineo ^ construere. 
secetur J3F in duas partes aequales in*£ [prop. 
X]^ et ducatur j4Ey et ad EF rectam et punctum in 
ea situm E angulo ^ aequalis construatur L PEZ 
[prop. XXni], et per A rectae EF parallela ducatur 
jiH [prop. XXXI], per F autem rectae EZ parallela 
^l ^ Z ff ducatur m, itaque parallelo- 

grammum est ZEm. et quo- 
niam BE^^EF, erit 
jr^S^ T ^ ABE = AEr, 

nam in aequalibus basibus sunt J3£, EFet in iisdem 
parallelis BF, AH [prop. XXXVIII]. itaque 

ABr = 2 AER 
uerum etiam ZEFH =2 AEF*^ nam basim eandem 
habent et in iisdem parallelis sunt [prop. XLI]. 
quare ZErH = ABr. et angulum FEZ dato angulo 
^ aequalem habet. 

Ergo dato triangulo ^JJFaequale parallelogram- 

m. 1; tajj ry Bp, PV m. 2. 9. TSftvsad^to p. xaxa to E 

8l%a F. xa/] om. 9. 11. TEZ] ZEF F. 12. tJ] om. 
F. ET^ om. F; mntat. in BF m. 2V. 13. EZ] ZE Bp, 
V m.2. TH] Htt.. r in ras. V. 14. hxlv PF. 16. 

laxl'] icxiv P, ^axat F. slatv P. 17. Post avxaig F habet 
Xomaig delet. punctis. xaig] insert. m. 2 F. BJPl corr. 
ex BEF P. 18. XQfyoivov] P, V m. 2; om. Theon (BFbp,V 
m. 1). 19. ZETH] F in F dnbinm est. 20. AET] 

ATE F. 21. kaxiv avTe5]^mg. m. 1 P. 22. iaxlv P. 

23. rEZ] TE e corr. m. 2 F. 24. xg J] xm J F. '26. 

Tc5 ABF] om. B, mg. m. rec. F; xm corr. ex to m. 1 b. 

7* 



100 STOrXEIilN a. 

AijAd/^afiftov GvviGxttZ(i.i ro ZEFH iv yavi^ zy i 
FEZ, ijtis ietlv ffft] xfj ^- oirep li«t itot^ffat. 

Tlavtos «ccifaHTiloyifiiiiiiov tmv nsf/l t-qvi 

5 diatietQov Tcapa^ktjXoypainKav tu iitt(faiii,THffJo-M 

(lata taa aAA^Aots iCrCv. 

"Eata na^KHTji.oyQanfiov to ABF^, Std^tQos Sh | 

^ffrra ta E&, ZH, ta di lEyofieva aaQanXTjpmiiara za 
10 BK, K^' liyia, ott l6ov istl to BK xaQaitXiJQfona 
ra KiJ «ttQanXiipd(iKrt. 

'Eitd yag itaQttlhjXoygttfiiiov iett lo ABFid, Std- 
fistQos Sh avtov Tj AF, feov ietl to ABF tpiyavov 
t0 AF^ ZQtymva. ndXtv, intl JtaQttXlrjloyQttfifiov 
15 iazt To E&, StdiiSTQog Si avtoii iettv 7] AK, Ceov 
ietl TTO AEK ZQiymvov za A@K ZQtymva. Sta ra 
avta Srj xal ro KZF tQtyavov tm KHV ietiv 
teov. insl Qvv to lisv AEK tQiyiovav ta A&K tQi- 
ymm ietlv teov, to Si KZF ta KHV, ro AEK 
20 tQiyavov iieta Toir KHT teov iatl tct A&K tQt- 
ymvip fisttt Toii KZ F' iezi Si xal ilov to 
ABP ZQiyavov oA§) Tp AAV teov' Xotnov «pa i 



XLin. BoetJus p. 384, 10. Apnd Proclum eicidit. 

1. vwinuxai] PBFbp; awiataxtti V; ««MffTOitjj qj. 
ZErH'^ Q corr. ip. h ■/atvia ip vith TEZ] om. F (mg. m. 
reo. h ymvia tb rno ZBT ^ 'hziv). 2. TEZ] seq. ras, 1 

litt. Pj ZEr B, V m. 2. ^hb] PVp; ^ BPb. Kot^ira(l 

m rae. p; Jei|ui P (f* £Ubi dii%ai mg. b). 3. JcafieTpo»' 

uvrcv p. 8. Post njji AV ia^V ra.. 1i add. Jta(»ttpo»i. 9. 

ZH^HZ F. nufiaKlnpiifiaita} -itlij^iiffiarii: ia ras. m. S V. 
Tu] m. rcc. P. 10. ttTiv P. 11. jiaQanlriQiiiian'] itaftt- 

sapra, V m. 2. 13. ij] foriv ^ F. raov] Tirov «(« F. 



I 



^ 



Fstructuiu eat ZEm in angulo FEZ, qui 
!gt angulo jd\ quod oportebat fieri. 



ELEMENTORUM LIBER I. 101 



XLIII. 



I 



I 



quouis parallelogrammo complementa paral- 
lelogrammorum circum diametrum positorum inter ae 
aequalia snnt. 

.A_S ._J Sit parallelogrammum ABV^, 

diametrus autem eius AF, et 
AF parallelogramma sint 
E9, ZH, et complementa, quae 
uocantur, BK, K^. dico, ease 
SK = K^. 
nam quouiam parallelogrammum est ABFA, dia- 
metruB autem eina AF, erit A ABr= AT/d [prop. 
XXXIYj. rursus quoniam parallelogrammum e&t E^, 
diametrua aatem eius AK, erit A AEK = A&K. 
eadem de cauaa etiam KZV -^^ KHF [ii.]. iam quo- 
niam A AEK = A&K et KZF = KHF, erit 
AEK + KHr-^ A&K + KZF [x. ivv. 2]. 



14, tetiv P. 16. £(=)] P m. 1, Bp, V m. 2; AKES P m. 

rec; JEXS F {AEK iu rM.), Vm. l,b, ZambertnB. I«»»j 
PFB; om. Vbp. raov a«a itniv P. 16, AKK'] AFE F; 
eorr. m ^XE m, 2. .J(9A:] 8K litt. in ma. V. rri «urii] 
toita BVb. 17. KZr~\ KHF p. A^HF] /CTZ p. 

Dein ftdd. le.ymvra P m. 2, FVbp. f«ov ^okV VU 18. 

^Eit] E litt. e corr. P. re/yuii.o»'] aupra m. 2 V. ^8JI] 
litt BK in raa. V. ipiymvm] om. p. 19. ieov liszl Vb. 

JCZri KHF p. XHr] Uit. H ems. F; Jtrz p. PoBt 

i6 ada. b o^ comp. m, 1. ^EA] £ litt. in raa. F. to 

AEK — 21. KZrj mg, m. 1 P. 20, ip/yaivin'] comp. Bupta 
m. 2 V. KHr} corr. ex AET m. 2 F. jWv Fp. iotiv 
fto» b. 22. Ajr\ litt, J e corr. F. 



102 



ET0IXEI2N 



fOov. 

IIccvTog apa naQttklrjloyqayi^^ov j;topiotJ tmv srepl 
T^i/ Sid(tbtQov TtaQttkh]XoYQtt^y.a>v r« naqazhiQd(ittTa | 

TlttQa t^v dod'£Fffai' «w^^sEav Tei dod-eVri rpt- 
^ini'^ ftfov n(i:()aAAi}AQ;jpap^oi' XKQa^aXetv iv 
Ty do&Eieti yoivCtt sv&vyQayifia. 

"EOTBt rj [liv do9ft6a Ev9eitt ^ jiB, to Se do&iv 
TQiymvov to F, tj iJe do&stifa yavia sv&vyQa{i(iog ij 
^' SeI: dii TtaQa ttjv do9tteav Ev&Etav tijv AR t^ 
Ao&fVTt TQtytova rra F taov 7taQai.lT]l6yQa[i[iov Jt«Q«- 
^aXstv Bv i'0T] T^ /} yaviq.. 

5 21vvt6TaTa t^ FTQiycovp fffov TtaQaXhjXoyQafifiov 
To BEZH iv ymvia TJj v^o EBH, ii iOTiv tOtj rj 
^" xal Jtttffftra StSTa iit sv&siag sivai tjjv BE ry 
AB, xal Sirix^oi j; ZH ial ro 0, xal dta roij ^i owo- 
rip^ Topv Sif, £^Z xaQaXXjjlos ^x*'" ^ '^®- ""^ ^^^~ 

gev2*(a ^ @B. jckI iitEl Eig naQaXX^Xovs zas A®, EZ 
EV&Eta EviaedEv rj &Z, aC ccQa vno A®Z, &ZE yta- 
viai dvalv oQ&atg elBiv l'aat. ai aQa v%o B&H, HZE 
S€o oQQ^mv iXaaaovEs Eieiv aC 8i ano iXaaeovmv ij 
dvo oQ&mv Eig anEiQOv dx§aXX6iievai avfixijttovatv 



XLiV. BoetiuB |), 381, H. 



1. iW lativ p. 3. jtojp^H oiQ. BVpi ofr. p. 100,4 
itaitfTQQV ttUTou p. 8. nai/apaleivl "■' ■- — ■- — « o 

iv'] iv ytavia, fl tfftiv i^eti Proclus; cfr. 
8'eMii'] mg. 'm, 1 F. 17. moi' V 
BH] Heq. ras. 1 Utt. F. ^* 

mg. m. 1 P. 20, 9B] B© F, 



CampauuH, I 

18. AB] A@ «. ... , 

AB F. xai— 30. eB].,i 

2t, cv&eias BVp. ^ J 



ELEMENTORUM LIBER I. 103 

uerum etiam ABF = A^F. itaque etiam 

BK = K^ [x. ivv. 3]. 
Ergo m quouis parallelogrammo complementa par- 
allelogrammorum circum diametrum positorum inter 
se aequalia sunt; quod erat demonstrandum. 

XLIV. 
Datae rectae parallelogrammum dato triangulo ae- 
quale adplicare in dato angulo rectilineo. 

Sit data recta AB, datus autem triangulus F, da- 
tus autem angulus rectilineus ^. oportet igitur datae 
rectae AB parallelogrammum dato triangulo F ae- 
quale adplicare in angulo aequali angulo ^. 

construatur parallelogrammum BEZH triangulo 

r aequale in angulo EBH, qui 
aequalis est angulo A [prop. XLII], 
et ponatur ita^ Mi BE, AB isi eadem 
» recta sint, et educatur ZH ad &, 
et per A utrique BH, EZ parallela 
ducatur A@ [prop. XXXI], et duca- 
tur SB. et quoniam in parallelas 
A@, EZ recta incidit ®Z, ^ 
S ^ A L^&Z + @ZE 

duobus rectis aequales erunt [prop. XXIX]. itaque 

LBSH+HZE 
duobus rectis minores erunt; quae autem ex angulis 
minoribuSyquam sunt duo recti, in infinitum producuntur, 

inBaBv] P; iftnsntayitsv Theon (BFVbp); cfr. p. 106,14. 108, 
26. aQa] om. P. AGZ] BHG j?\ corr. m. rec. 9ZE 

— 22. B9H] mg. m. rec. p. 22. siatv taat] PBF; taai 

slaCv Vbp. Ante af insert. comp. xat B. £0Z, SZE 

P. 23. ano] an p. 24. hY.^aXXo\ii.svat sig ansiqov p. 

i%§aX6fi,Evai, P. 





104 STOIXEKN «'. 

ai&B, ZEaga ix^alloiievui ovfi^tEaovvrat. ixpEpArf 
a&<aSttv xnl OvfintnTiTaaav xara ro K, xal Sia xov 
K oi^fieiov onoriQa tmv EA, Z® jtaQai.k7ii.og ^%%a} 
ij KA, x«l ix^B^lie&BKSKV at @A, HB iltl ta A, M 
6 OijnEta. ^aifttlli]l6yQa[i(iov apa iorl ro ®AKZ, Sta- 
^crpos de avrov 7] @K, stegl di rijv &K TtaffaXli]- 
loygafifta ftev ta AH, ME, ta di kayo^Bva itaQa- 
7TkrjQm[iata ta AB, BZ' Isov «p« iatl ro AB ip 
BZ. akXa to BZ ra F rpiydva ietlv i'aov xal to 

10 AB Squ ta r iariv Heov. xal ijitl Catj iatlv ^ va6 
HBE yavia tfj v:t6 ABM, aXXa tj vxb HBE rp z/ 
ietLv iari, nal ■^ vsto ABM «p« t^ ^ yavia iatlv larj. 

HttQtt T^v do&eiettv aga ev&ttav rijv AB ta i 
9ivti t^iymva tp F teov 3r«p«AiijAdj'paftftoi' atapai 

16 pXtjtat ro AB iv ytovC^ tfj vno ABM, ^ iotiv taii 
t£ ^" orecp ISei xoiijeai. 

^^: 

Ta do&ivti Ev&vy^diina toov naqaXXvik^- 
/paftfiov aveti^eae&ai iv r^ tfodftffjj ymviaev- 
20 ftvyffafifip. 

"EffTCJ ro ftiv So&lv sv&vygafiiiov ta ABPzl, ^ 3h 
So&etaa yavia f^#i5ypafi(iog tj E' Sil li^ ta ABFjd 
tv&vyQtt(tii^ Heov 3taQaXX^X6ysfa[i[iov evar^eaa9ai. iv 
ty So&tCeti yavCa tj^ E. 
26 ^EntlEvx^o) 1} ^B, xal evvsatato} rp AB.d rpi- 
ytovp toov sai/aXli^X6yQa[i[iov x6 Z& iv tfj vao &KZ 
XLV. Boetius p. 384, 17. 



1. 6B] AB K. 4. ix^E^I^a#(o ip. HB] ue <p. 

M\ aeq. lacuna 3 litt. <p- 6. {azlv PF. SAKZ] e «uir. 

r. 6. 9K] (prius) BH rp. ie] supra m, 2 F. 7. fli 

IfyofiJiia] oij (iB 9, seq, fie»ii enan. m. 1. 8. ra\ om. B. 

i<stCv P. 9. oUa «al to V. 10. ^BJ coiT. ei JB lo. 2 P. 



1 

I 
I 



ELEMENTORUM LIBER I. 105 

ooncurrunt [ah. 5]. itaque &B, ZE productae concur- 
rent. producantur et concurrant in Ky et per K punctum 
ntrique Eji, Z6 parallela ducatur KAy et producan- 
tur &A, HB ad puncta jd, M. itaque 0jiKZ paral- 
lelogrammum est, diametrus autem eius BKy et circum 
SK parallelogramma AHy ME^ complementa autem; 
quae uocantur, AB, BZ, itaque erit AB ^ BZ [prop. 
XLni]. uerum BZ ^ F, quare etiam AB — F [«• 
ivv. 1]. et quoniam L HBE^ ABM [prop. XV], 
nerum L HBE — A^ erit etiam L ABM= ^. 

Ergo datae rectae AB parallelogrammum AB dato 
triangulo F aequale adplicatum est in angulo ABM, 
qui ato angulo A aequalis est; quod oportebat fieri. 

XLV. 
Datae figurae rectilineae aequale parallelogram- 
mum construere in dato angulo rectilineo. 

Sit data figura rectilinea ABF^i, 
datus autem angulus rectilineus E, 
oportet igitur figurae rectilineae^Jjr^ 
aequale parallelogrammum construere 
in dato angulo K 

ducatur AB^ et triangulo ABA 

aequale construatur parallelogram- 

jf ^ jiT mum Z0 in angulo @KZy qui ae- 

ta] To F. insf\ del. August. 11. HBEVlitt. H in ras. 

m. 1 B. dn' F. 12. ABM] in ras. m.2 V. a^a] om. 




B; mg. m. 2 V. yoavta] om. p. 13. iaxiv] om. -tp. 16. 

~ I. 1 P. Tr7] bis qp. 

^f^ffj] dcfj Bp. 26. imisvyvvcd^to FVb (in b enpra scr. m. 1 



— » O • jj JT J ~ — — » 

To -«4B iv ymvici t^] nig. m. 1 P. tw] bis qp. 24. t^ ^o- 



« Z)- ^] yap ^ P- ^^] mutat. in B^ m. 2 V; AT P, 

mg. y^. xai ij z/B. ABJ] BA supra scripto -J F; ^lBr P. 
Tpiycoyo)] svd^ F, seq. yQafifnov 9. T^tyflcVo) ^<rov] corr. 

m. 1 ex x^iytovov tco P. 



106 STOIXEliiff a 



yavCa, ^ ttfiii' fffij t!j E' xal jtaQa^e^X^e&o} napor 
zifV H& Ev^tiav Eco ^BT Tptj^oivp fdov nuQalljjlo- 
■yifay,{tav ro HM iv ty vno H&M ytavia, ^ ieriv 
tat} r^ E. xa\ iitsi ij £ yiavCa ixariQa zwv vno &KZ. 
5 H®M iariv fffi?, xal ^ vxo &KZ aga tjj vith H&M 
ieriv fojj. xotvTi jtQoexEie&a rj vao K&H' ai uQa 
irao ZK@, K&H raig vno K&H, H&M taai aiaiv. 
&U' aC vxo ZK&, K&H Svclv oQd-aig leai. deCv 
xal aC vao K&H, H&M a^a Svo og&ais leai sC- 

10 eCv. apog St} xivi sv&sia tjj H& xal ra wpog avrjj 
OijfiECp za @ dvo £v9£i:ai aC K&, @M fii] iitl 
avza fiiffTj xsCfiBvai rag e^jeI^? yavCag dvo oQQatg 
i'a«S jcoLovaiv in Ev&sCaq apa iazlv tj K® rij @M' 
xal ijisl tig wapa^AjJioug zag KM, ZH tv&eta iv- 

15 imetv ^ &H, aC iva}.i.ai yavCai at vno M&H, &HZ 
taai aXXrikaiq eiaCv. xoivij itqoaxhCo&a ij vno &HA 
at aqa vito M&H, ®HA ratg vno &HZ, &HA teai 
eCeCv. mi' at vxo M&H, @HA Hvo 6p*«rs taat 
eieiv xal at vno @HZ, @HA aga Svo op&ats 

80 teai eCeiv la' Bv&eias apa iorlv tj ZH t^ HA. 
xal iatl ij ZK i^ ®H fffij re xal ^iapaAAij^ds ieziv, 
aXXa xal ri @H zij MA, xal ^ KZ aga ty MA fffi/ 
TE xal stapaXliilog iariv xal im^svyvvovaiv avzas 
tv^stat aC KM, ZA- x«l aC KM, ZA ai/a taai « 



1 

IV V 



1. yuvia] mg. m. 1 P. (■irpi isTiy P. 3. if»] &H P. 
fv9tlav] corr. ei iv0ttU F. AjrV, laij hrtv p. 

HeU] H anpra F. 7. tlatv teai V. 8. alla PB. St 
viv] ivo F{ Gorr. m. 2. taai tlaiv] ilan' idtti p; teai ttoC 

Vb. 9. 3va] P, F m, 1; Svaiy BVbp, F m, 2. tlaiv] '" 
V; comp. b. 11. K9] 9K P. 12, ivaii' BVbp. 
BM] 6 corr. »,8 P. 14. ZH] ZK ip; ZA p; H in raa. i 
V. Bv&tias P. Sapra ^►EireoE»' in F 8cr. ^^ji^uiidii* 

16. eloiv] PF; ciai uulgo. 17. Post afa raa. 1 Urt. F, 



1 



^M ELEMENTORUM LIBER I. 107 

qualis ait angulo E [prop. XLIIj. et rectae H& par- 
allelogrammnm HM triangulo ^BF aequals adpU- 
cetur in angulo H@M, qui aequalis sit angulo E 
[prop. XLIV]. et quoniam angulus E utrique ®KZ, 
H&M aequalis eat, erit etiam L&KZ =H@M[k. 
iw. 1]. communia adiiciatur L K&H. itaque ZKS 
+ K&H = K&H + H&M. uerum ZK"© + K@H 
duobuB rectis aequales sunt [prop. XXIX]. itaque etiam 
K&H-\- H®Mduobus rectis aequales sunt [x. lvv.2]. 
itaque ad reetam quandam H& et puuctum eius & 
duae rectae K&, &M non in eadem parte positae an- 
guIoB deinceps positos duobua rectis aequales efficiunt; 
in eadem igitur sunt recta K& et &M [prop. XIV]. 
et quoniam in parallelas KM, ZH recta incidit &H, 
anguli altemi M&H, &HZ inter se aequales sunt 
[pr^. XXIX]. communia adiiciatur /. &HA. itaque 
M&H-{- ®UA = &HZ + &HA [x. Ivv. 2]. uerum 
M&H + &HA duobua rectis aequalea auDt [prop. 
XXIX]. itaque etiam &HZ + &HA duobus rectiB 
aequalea sunt {x. Svv. 1]. quare ZH, HA in eadem 
snnt recta [prop. XIV]. et quoniam ZK rectae &H 
aequalis et parallela est [prop. XXXIVj, uerum etiam 
SH rectae MA [id.], etiam KZ rectae MA aequalis 
et parallela est. et comungunt eas rectae KM, ZA. 

MBH\ e e corr. V. QHA] e corr. F. SHZJ e corr. V; 
»HA P. BHA] enz P, ilaiv fuBt p, foffi] ftij ip (ftfai 
P). IS.bUoPB. M0H] litt.©Hin raa, b. ffuoiVBVbp. 

19. elai V, connj. b. xoi 0^—20. ilaiv] mg. m. 1 BF. 
agv} om. Fb; ma. ni, 2 V. 3vo] P, dvaiv uulgo, 20, tlmv 
fyat p. iativ] iOTlv xal P, 21. ZK] KZ P. 22, i, &H\ m 

om. F; corr, ex j] ES m, 3 V, %al ^ KZ apa tj MA] om. M 

b. 23. imiv] iffit BV. 24. aea] bp, et V fli>d pnnctis ■ 

. delet.) coni, Aogust n p. 311; om. PBF, ■ 



L 



108 STOIXEIHN tt', 

xttl wap«A^*)Aoi tlOLV ffcpft/ATjAoypaftfiOv a(fa itnl to 
KZAM. xul inBi HeQv ierl t6 fttv ABjJ tgiycavov tp 
Z& n<xQCilkr]XoyQ«(i,fta f ro tfi ^ST ra HM, oAov 
«pa t6 jiBr^ Evd^Qafiiiov oAp ra KZ^M KorpKl- 

5 AljAof^piIft^G) ^ffTlv ftfOV. 

Tp «pa do&ivtt. sv&vyQdn(ia rw ABF^ iOov Kap- 
«AAijAoVpaftftoi' ewiezattti ro KZAM iv ymvia tjj 
tijro Zff^M, ^ ^tfru' fffij tjj do&eioti t^ E' on«p f&i 

K0£^Cff£. 
10 fiS'. 

'.^ao TJjs So&siaijs tvd^tias rtTpayatvov ava- 
y^a^tti. 

"Earm ri So9Btea £v&eta ij AB " det 5^ aitv r^g 
AB EvffEfai TEZfjaytavov dvay^^ai. 

15 "H%%m rij AB ivd^tia «wo tou «p6s kut^ erniiiov 
Toii A jrpog op#«g ^ -^f'. xal xBie&ei i^ ^B fefjj ^ 
^ii' x«i 6ia (iiv Tou ,^ atjfisiov tfj >iB napa/AijAog 
^j;3'(U ^ .rJE, Sitt dh Tow B eTifiBiov rfj A^ xaQttllij- 
iog ^x&a ij BE. HRpaXXrikoyQafmov uffa iaxl to 

20 A^EB- teri apa iarlv ij (thv AB r^ A E, ij di A^ 
ry BE. ttkka ^ AB rfj Ad ieriv teri' al tiaettpcg 
Kp« ttH BA, AJ, ^E, EB faai aX^Xais eleiv leo- 
jtXivtfOv aqa iatl ro AAEB jraptrAAijAoypafiftov. liyet 
d^) ort xal offS^oytoviov. insl yap sig irapaAAijAovg 

26 zag AB, ^E tv&Bitt iviitBaav i] AA, at apa ino 
BA^, A^E ytoviai Svo op&ats laai slaiv. 

XLVI. AtnmODiiiH in Porplijr. fo!. 48V BoetioB p. 384, 19. 1 

1. thivl PFp; tlei Qolgo. Seq, raa. 2 litt. F- ltni'\ 1 

lotlv FV. 2. %al— fti»] mg. m. 1 P.l ABd'] AJB i " 

ABF P, et F, corr. m. rec. 3. zJEr] dAT P. 5. ^diI 

lao*\ PFpi teov iarir V; taov lexC B et comp. b. 6. ip] i 



ELEMENTORUM LIBER L 109 

quare etiam KM^ ZA aequales et parallelae sunt [ae. 

iw. 1; prop. XXX]. parallelogrammum igitur est 

KZAM. et quoniam A ABJ = Z©, ABF^^HM, 

erit ABFJ = KZAM [x. iw, 2]. 

Ei^o datae figurae rectilineae ABFA aequale 

parallelogrammum constructum est KZAMin angulo 

ZKMy qm dato angulo E aequalis est; quod oporte- 

bat fieri. 

XLVI. 

In data recta quadratum construere. 

Sit data recta AB, oportet igitur in recta AB 

quadratum construere. . 

ducatur ad rectam AB a puncto in ea sito A per- 

pendicularis AF [prop. XI], et ponatur AA = AB 

[prop. II]. et per punctum ^ rectae AB parallela 

ducatur zi£^ per B autem punctum rectae AA paral- 

j, lela ducatur BE [prop. XXXI]. paral- 

lelogrammum igitur est AAEB. itaque 

AB = AE et ^^ = 5E[prop. XXXIV]. 

^ uerum AB = AA. ergo 

BA = A^ = AE = EB [x. ivv, 1]. 

quare aequilaterum est parallelogrammum 

A^EB. dicO; idem rectangulum esse. nam 



^ ^ quoniam in parallelas AB,^E recta in- 

cidit^^, BAA -[- ^^^ duobus rectis aequales sunt 

(alt.) corr. ex to m. 1 b. 7. avviazatcct FVp. t6] corr. 

ex Tj m. rec. P. 8. tj] (alt.) om. b. 9. iv alXm Sst^ai 

mg. m. 1 b. 12. Post prius rj ras. p. 16. 17I (ait.) corr. 

ex XV V. 18. dE] corr. ex ^E m.2 p. 19. iativ P. 
21. aXXd] dXX' F; dXXcc %ai Vb. 24. *?)] ii Vb; om. P {pi 
supra comp. m. 2). 25. ev-a-atag V, svd-Biug V m. 2 et b. 
r/] T^ 9. Post aQcc lacun. 3 litt. 9. 26. BAJ] litt. BA 

in ras. m. 1 B. AJE] litt. JE e corr. F. Svaiv BVbp. 



110 ETOIXEmN «'. ' 

di ij vao BA^' opft^ UQtt xttl f, vao A^E. rav &% 
jiaQaXk^XoyQKfHKOv %a^Ciav ai ditEvavtiov nX^VQaC ts 
naX yiovCat teai «AAi/Aaig eieCv opflTj «pa xai sxutbqk 
rmv daevavTLOV Tciv imo ABE, BEA yaviSv opOo- 
5 yeivtov a^a iozl z6 AAEB. Mftj;*») Se xal Cso- 

ItXFVQOV. 

TeTQayaivov aga itSTCv xaC ioxiv dnb T^q AB ev- 
QtCas dvttyeytfafifievov " ojtep i'Su jioiijeai. 



10 'Ev Tots og&oycavCois T^iydvois t6 ttno t-^g 

Tijv OQd-ijv ytovCav vnoxtivovarig nXsvQag r, 

TQayavav Heov iffTl rofg kko Tmv zrjv OQ&i^ 

ytavCav asQisxovamv 3t?.EVQmv tETpaycivoig. 

"Eexa T^Cymvov OQ&oydvtov zb ABF op^v ^xov 

15 TTjv vnb BAF ymvCav' ^iyin, OTt lo ajto t^s BF tb- 
TQayavov laov isTl totg dno ttav B/i, AF TtTpayto- 
voig. 

'AvttyEyQixq}&a yuff nito ftiv Tijs BF TETgdycovov^ 
xoBJEr, dno dl tAv BA,ArTttHB, &r, xal dieci 

ao Totr A bxoTSQa tav BA, FE aaQdi,Xr]Kog ^j^d-ca rj A^'- 
xal iitE^svx&iaaav al A ^ ., ZF. xnl daEl opfl-i} iativ 1 
Sxaxifftt xav vnb SAF, BAH yaviiav, repog d^ i 
BV&ECa TTJ BA xttl T(5 ^QOS ttvrij Sii(iBCa t^ A Svom 
EvftEltti ttl AF, AH ft^ ijtl ta avxa fiipi} xsCfievai 

25 Tag i^s^ijs yavCag 6va\v OQ&alg Csag TfoiovStv' ia 4 
EV&ECag ttQtt iatlv ^ PA t^ AH. Sia xd avxu S^ xaLu 



SLVU. Pappns Ip. 178,11. Schol. in ArcHm. III p. 388.j 
BoetiuB p, 38*, 21. 



1. »b(] maert. m. rec. b (comp,), 5, ivTlv FV; comp. ^M 



ELEMENTORUM LIBER I. 111 

[prop. XXEK]. uerum L BA/I rectus esi itaque etiam 
L A^B rectus. sed in spatiis parallelogrammis latera 
anguliqae opposita inter se aequalia sunt [prop. XXXIV]. 
itaque etiam uterque angulus oppositus ABEy BEA 
rectus est. rectangulum igitur est AAEB, demonstra- 
tom autem est, idem aequilaterum esse. ergo quadra- 
tom est [def. 22]. et in recta AB constructum est; 
quod oportebat fieri. 

XLVII. 

In triangulis rectangulis quadratum in latere sub 
recto angulo subtendenti constructum aequale est 
quadratis in lateribus rectum angulum comprehenden- 
tibus constructis. 

Sit triangulus rectangulus ABF rectum habens 
L BAT. dico, esse JJF* = BA^ + Ar\ 

construatur enim in BF quadratum BAEF^ in 
BAy AF uero HB^ BF [prop. XLVI], et per A utri- 
que BAy FE parallela ducatur AA [prop. XXXI]; et 
ducantur AA^ ZF. et quoniam rectus est uterque an- 
gulus BAFjBAHf ad rectam quandam BA et punc- 
tum in ea situm A duae rectae AF, AH non in ea- 
dem parte positae angulos deinceps positos duobus 
rectis aequales efficiunt; itaque in eadem recta sunt 
FAy AH [prop. XTV]. eadem igitur de causa etiam 

TO AJEB] mg. m. 2 V; in F snpra E scr. H. 7. BOt^v] 

(prius) PF; I<ft/ nulgo. 12. triv] ««pl rriv Ptoclus. 13. 

nBQitxovcmv] om. Proclus. 15. BAF] corr. ex BFA m. 2 F. 

Ante BF eras. A P. 16. taov] supra m. 2 (comp.) F. 

hx^v P. BA] AB F. 18. fiiv] om. F. 19. BFJE F. 
HB] corr. ex ££r m. 2 F. GF] F in ras. est in F; seq. in 
y m.2: xBTQdymva. 20. rjx^m nuQoiXXrilog p. AJ] J in 

ras. P m. 1. 23. BA] AB -p. 26. ta avtd] tavta Bp. 



112 STOIXEIiiN «'. 

^ BA rfi A® iexiv tJi' ev&iias. xal intl (arj iiJtIv 

17 ino ^BF yavia t^ vao ZBA' dp^ij yaQ ixazEQa- 

xotvi) TiQoaxECaQa »} v%o ABF' oltj apa tj vno ^BA 

oAfj r^ vnb ZBF iaztv iarj. xal iitel lOt} iozlv ^ 

6 (ilv AB Tji Br, j; 6s ZB rrj BA, dvo 8ij al J B, 

BA dvo TaFg ZB, BF ieat Eialv ixariQa ixatitf^' 

xal ycovia 1) VTtb /dBA yiovla ry vno ZBF ia^' 

(idats «9« ^ -^^ fiaasi t^ ZT [iativ'] Csvj, xal t6 

AB/} tpiyavov xa ZBF tQiydva iatlv iaov »a£ 

10 {iaztl tov (liv AB^ tqiydvov Stnkdaiov toBA TcaQ- 

alkijkoyQafi.fioi' ' ^daiv te yap f^v avtiiv i%ovSL trjv 

Bjd xal iv tats avtats *'"* roapBAXijioig tats B^, 

AA- Tow S^ ZBF tQiy(6vov Sialdaiov tb HB tetpd- 

ymvov ^daiv te yaq ndhv f^v avt^v l^ovat tijv 

15 ZB xal iv Tffifg uvtaTs siat naQttXX^riXots taZq ZB, HV. 

[ra b% tav iaatv Sntldaia iaa «AAijAoi; iariv~\ taov 

apa iatl xal tb BA %aQttX}.t}k6yQa^(iov Tp HB rs- 

TQttytava. ojxotrog ij^ im.^evyvv(iBveyv tmv AE, BK 

Set^&^^aBtai «al to FA izaQaXXijXoyQttfiftov taov t^ 

20 &r tetQttywv^' olov ttQtt ro B .J EF tBtQdyavov Sval 

^M tois HB, &r TEtQayoivois taov iativ. xai iatt to (liv 

^H BAEP tsTQaytovov dnb Tijg BP dvayQttfiv, td di 

^P HB, &r aab Toiv BA, AF. ro «pa dnb z^s Brnktv- 



I 

I 



1. W tv^tias iarir V. 2. ^BT] j^TS F; corr. m 
4. ZBT] litt. r e corr. F. iariv i'an] fffij laxiv p. 
htlv n p.\v JB xji Sr ii Si ZB tiBA] P; om. ThBOn ( 
Vbp). 6. ei] P; om. Theon (BFVbp). JB, BA] in : 
m. 2 Vi AB. BA F, corr. m, 3; AB, BJ b. 6. «vai B 
evtiv V. BZ, BF BFp, V m. a. 7. ZBT] litt. ZB"a | 

corr. p. foi) iati V. 8. lativ tait] ftij F; tvn iatl ' 

%tti] comp. supra. m. 1 b. 9. ABd]AdB F. fooj 

V. _ 10. ^otijom. P. BA] BJ F, et b, corr. m. 

11, aui^ rfiv avxriy ixet p. ^ooani P. 117»] oorr. 



^ 



ELEMENTORUM LIBEB I. 



113 



BA, ji0 in eadem recta sunt [prop.XIY]. et quoniam 

L ^Br= ZBA (nam uterque 
rectus est); communis adiiciatur 
5jr LABR itaque 

L dBA^ZBr\%Jvv.2\ 
et quoniam AB — BF^ 

ZB = BA [def. 22], 
duae rectae ^ByBA duabus ZB^ 
BF aequales sunt altera alteri; 
et L ^BA = ZBr. itaque 
AA = Zr, A ABA = ZBF [prop. IV]. et 

BA = 2ABA', 
nam eandem basim habent JS^ et in iisdem parallelis 
sunt BAj AA [prop. XLI]. et HB = 2 ZJSF; nam 
rursus eandem basim habent ZE et in iisdem sunt 
parallelis ZB^ HF. itaque^) BA = HB. similiter 
ductis rectis AE^ BK demonstrabimus, esse etiam 
FA = ®r. itaque BAEF = HB + &r [x. ivv. 2]. 
et^^^Fin BF constructum est, HB^ ©Fautem 




J A 



1) Ex comm. concept. 2; nam uerba ra 6\ tcov tcoav Si' 
nldaicc tacc dXXi^Xoig iaxlv lin. 16 cum x. |yv. 5 interpolata 
sunt; c&. p. 91 not. 1. 



m. 2 F. 12. tlai\ iatt p. BJ,AA tov] mg. m. 1 P. 

13. HB] BH P. tBTQdyaivov] comp. b; supra hoc uerbum 

in F scr. nciQalXriXoyQafifiov m. rec; item lin. 17 et 20. 14. 
yap] ydff avzm p. I^jovfftl l^jovtfti' PF; ^3;«* p. 15. ZBl 

BZ p. stai] satt p; om. V; siaiv F; comp. b. 16. iat^v] 
hlaCv V. 17. iatCv P. 18. di}] m. 2 P. 19. FA] AA, 

ut uidetur, F; corr. m. 2; ^r V, corr. m. 2. 20. BdET] 

z/EBr p. SvaCv P. 21. Caov iatCv] PF, comp. b; iatlv 
taov p; faov iatC uulgo. naC iativ P. 22. -JEBF p. 
ayayeypof^ seq. ras. 2 litt. F, dvaysYQafifiivov p. ra] supra 
F. 23. Ante HB ras. 1 litt. F. Ante BA ras. 2—3 litt. F. 
BA] BJ (p {BA F). 

Euclides, edd. Heiberg et Monge. 8 



114 ETOIXEIUN a. 

pKj TiTQayavov teov IotX rofg axh rav BA, ArjiKsV' I 

Qtov T£TQaytovoig, 

'Ev Kpa Toig OQ&oyavtotg TQi,ymvois tb dao r^g I 

iijy op&ijv ytoviav vitoTeivovG^^s ^^evgag tBtpdyavov T 

5 ieov iOTl Tofff (Jroo teiw tijv o^^fr^v [yoviav^ wfptfjjou- | 

Uciv jtAiuprav Tfrpaytiivots' o^bq ^Ssi dst^ai. 

'Ekv tptyavov ro axb [ttag Trov jrAjup. 
rEtpayovow fooi' jj TOts anb twv koijcav t 

10 TQtywvov ffvo Jtifupdjj' Tcrpa/wvoiff, ^ JtfptExo- 
(tfvjj ytavia v-ko Ttof AotJtrav rotJ rptyrawow ffuo 
jt^Erpoiv opd'?} latLV. 

Tgiyiovov yccQ xov ABF to aJio ^tcg z^s 
TtlsvQas TtrQayavov fffov ^OTco Tofs ajro tmv BA, AP 

15 nXfVQiav Tttpayiovots' kiym, otl opdi] iOTiv ■»} vxo 
BAF yavla. 

"Hx^^ta yap «wo tou A ffijftetou rfj AF sv&eCa Jtpos 
op^Kj ^ ^^ x«i xato&m r^ BA t^eri ^ ^z/, xai ijte- 
gtit^ftia 17 z/r. ^jifl itJri iBTiv 17 z/^i Tf/ AB, iaav 

20 ^orl M(l t6 ojto t^^s .</^ TCTpaycjjioii Tp aJio t^g ^B 
TfTpaytavra. xotvhv Jtpoffxefff^o t6 tJjti T^g AT Te- 
Tpayojfov TK apa ajro xav AA, AF rtrQuyava TtfK 
tOrl TOig a3r6 Troi' BA, AF terQaymvots. aXla Tofs 
(ifv ajto xav dA, AF taov iorl ro ajto tijs AP' opfr^ 

25 yag iariv 17 vzb AAF yatvia' Tofg di axo rav BA^ 
AT laov ietl t6 aito t^g BF' vxoxsiTat ya(f to api 

SLVUI. Boetiua p. 381, 26. 

1. lott* ftio* F. ^ ^DiC» P. Bv*] BJ 91. _ 3. iv] 

F; COIT. m. rec. af^ayavois p. 4. intitivovffi/; V; < 



(0- 

ivom 



ELEMENTORUM UBER I. 115 

in Bji, AF. itaqne qnadratnm lateris BF aeqnale est 
qnadratis latemm BAy AF. 

Ergo in triangnlis rectangnlis qnadratnm in latere 
snb recto angnlo snbtendenti constmctnm aeqnale est 
qnadratis in lateribns rectnm angnlnm comprehenden- 
tibns constmctis; qnod erat demonstrandnm. 

XLVm. 

Si in triangnlo qnadratnm nnins lateris aeqnale 
est qnadratis reliqnomm dnomm laternm triangnli, an- 
gnlns reliqnis dnobns lateribns triangnli comprehensns 
rectns est. 

nam in triangnlo ABF sit BF^ = BA^ + AFK 
dico, LBAF rectnm esse. 

dncatnr enim a pnncto A ad rectam AF perpen- 

dicnlaris A^ [prop. XI], et ponatnr AJ = BA, et 

jf dncatnr z/r. iam qnoniam /tA = AB, 

erit^) etiam JA^ = AB^. commune ad- 

iiciatur AF^, itaqne 

^A^ + Ar^ = BA^ + An [x. ivv. 2]. 
uemm ^r^ = AA^ + ^F*; nam L ^AF 
rectus est [prop. XLVII]; et BF^ = B^« 
A Zi 5+^r^; hoc enim suppositum est. itaque 




1) Hoc ex definitione quadrati (22) sequitur. 



m. 1. 6. laxCv PF. ymviav] om. PBF. 12. htiv^ 

PFV, Proclus, comp. b; iaxi Bp. 15. Post nX^vgaiv ras. 

5—6 litt. b. 19. z^r] ^ in ras. b. ^««/] PBVb; ^nhl 

ovv Fp; xal insi P m. rec. iativ] comp. supra m. 2 F. 
AJ F. 20. iativ P. to] supra m. 1 b. ^B] B^ p. 

21. xoirn B. 23. iatCv P. AT] om. 9. 24. iatCv P. 
^r] ^r tBXQdyaivov p. 25. FAJ F. BA] AB B. 26. 
iativ P. vnoTifitai 9, seq. tai m. 1. 

8* 



116 STOIXEKiN «'. 

UTcb r^s ^jr tstQoiycDVOV l6ov i6tl tp octco tijg BF 
tstQaydvoD' &6ts Tcal TclsvQa ^ j^Ftfj BF i6tiv t6ri' 
xal iitsl^t0ri i6tlv ii /dA ty AB^ %OLvri 81 r/ AF^ 
8vo Sri at JA, AF Ho tatg BA, AF t^ai slaCv 

5 xal §d6vg ii /JF fid6si tfj BF t6ri' ycovia aga ^ vno 
AAFyovtcc tfj vno BAr[i6tLv] t^rj. OQ^ij dh ^ vno 
AAF' ogd-ri a^a xal ^ vtcv BAF, 

^Eav aQa tQiycivov t6 ccjco fiL&g tciv xIsvqAv te- 
tQciycovov t6ov y totg dico tmv loiJCciv tov tQLycivov 

10 8vo TcXsvQciv tstQayoivovg, fi JCSQvsxoiidvri yoDvCa vtco 
t&v koiit&v tov tQiycivov Svo nksvQmv OQ^rj *i6tiv* 
onsQ SSsi Ssti^av. 

1. iexiv P. Tco] to b; corr. m. 2. 4. ^77] absumptum 
ob pergam. ruptum in F. dvaC BVbp, F m. 2. %lciv\ 

PF; comp. b; Blisi uulgo. 6. tJJ 17 9. tcri\ PBbp; tcri 
iaxCv F ; Tcnj iaxC V, sed iatC punctis del. m. 2. ^] supra P. 

vno] om. P. 6. ictiv] BFVbp; om. P. 8. xQi^ymvm p. 

10. In mQtsxo(iivrj ante x ^s. 1 litt. b. yoivCoc om. p. 
In fine: EvhXsCSov axotxsCmv a' PB; EvvXbCSov ctoixBicov xrig 

®ecovos sndoOBmg § F. 



ELEMENTORUM LIBER I. 117 

z/r* = Br^ [x. ivv. 1]. 

quare etiam ^r=Br. et quoniam ^A = ABy et 
communis est AF^ duae rectae ^A, AF duabus BA^ 
AF aequales sunt; et basis z/F basi BF aequalis 
est. itaque L ^AF^BAF [prop.VIII]. sed L ^AF 
rectus est. itaque etiam L BAF rectus. 

Ergo si in triangulo quadratum unius lateris ae- 
quale est quadratis reliquorum duorum laterum tri- 
anguli^ angulus reliquis duobus lateribus trianguli 
comprehensus rectus est; quod erat demonstrandum. 



a. n«v aa(f«i,lT]l6yQaiijiov 6Q^oyt6vi.ov aa^t- 
i%Ea9tti liytTtti vab dvo rmv rijv dp&^v ycaviav «bqi^ 
iX^^vemv EV&Eimv. 
5 |3'. navzbg 6i attpaK^TjloyQaiijiov ^agCov tiovI 
«eq\ t^v dicmiVQOv avzov naQttllt]loygKii[iav Svt 
oaoiovovv aiiv rotg dvol aapanlriQmficcai yvmfitov zb-J 

} 'Eav mai Svo Ev&Ettti., Tfir}&^ Si ^ ETiQn^ 
ttvzmv eig oSttdijjiovovv rftijpara, ro iTEpt- I 
sx6ftfvov oQ&oymviov vao rav Svo Evd^eiav 
iaov iatl zotg vjto ze T^g az(irizov xal ixa- 
0ZOV Tiov zfttiitttzmv ntQiExofiivoig opd^oyiaviots. 

j "EozaOav dio Evd^etat at A, BT, x«i rErfMjffdra ij 
BV, ras hvjiEv, xark tcc ^, E griftEtK' liya, ozt zo 
vitb zav A, BF xeQiExofiivov OQ&oytoviov tSov iezX 
ta TE vTib tAv A, B/I TtEQtexoiiiva oQd^oyavip x«l 
Tta V7tb ziDV A, j^E xal izt rta vxb rmv A, EF. 



Def. 1. Heto def. 57. Boetiua p, 378,8. Def. 2. Heto C 

def. 58. Proolus in Tim. 83d. Boetiiis p. 373,11. Prop. L 1 

EutociuB in Archim. 111 p. 40, 29. 266, 7. Boetiua p. 385, 4. 

EvtdiiSov aiaixtiiav dtvteQov B; E-arlBiSov ett T^g Si»i 
vot iiSoaeuis aiot%eiia¥ itVTCeov V; BvKliiSov aioiieimv v~~ 



n. 

Definitiones. 

1. Quoduis paraUelogrammiim rectangulum com- 
prehendi dicitur duabus rectis rectum angulum com- 
prehendentibus. 

2. In quouis autem parallelogrammo spatio utrum- 
uis parallelogrammorum circum diametrum positorum 
cum duobus supplementis gnomon uocetur. 

L 

Si sunt duae rectae, et altera earum in quotlibet 
partes secatur, rectangulum duabus rectis comprehen- 
sum aequale est rectangulis recta non secta et sin- 
gulis partibus comprehensis.^ 

Sint duae rectae A^ BFy et secetur jBJT utcumque 
in punctis -^, E, dico, esse 

AxBr=AxB/l + AxJE+AxEr. 



1) Arithmetice ax(6 + c + (^ = ah + ac •^- ad. 

Gscovog snSoaatog § F. 1. 0901] om. P[BF. Numeros om. 
PBF. 10. idv] seq. ras. 2 litt. F. maLV B. 13. ictCv 

P. totg'] corr. ex teo P. vno t«] ts vao P, rs dno F. 
14. nsQisxofisvotg 6Q9'OY<oviotg'] corr. ex nsQLSxofisvtp o^oyco- 
vtfi) P. 16. hvxfv] PBF; itvxs Vp. ari(tsVa] supra m. 2 
V.' To] in ras. V. 17. istiv P. 18. reo] in ras. V. 
ts vno] PF; vno V; vno ts Bp. 19. tmv] PVp; F insert. 

m. 2; om. B, F m. 1. iti] om. P. tm] corr. ex tmv V. 



120 ETOIXEiaH |J'. 

"Hx&a yag ano Toii B r^ BF jrpos 6p&as 17 jBZ, 
«tei xsia&w Tf/ ^ fffTj ^ BH, xal Sia filv roii H r^ 
Br kkpk;IAi?Aos iJx*'^ V ^®i *'« ^^ iw*' ■^■! E, ^Tfl 
BjH SK^KAAijAot ^x^^^^'" "^ ^^j -'^^j ^®- 
5 "laov *jj f'Oi( ro B& roEs B-K", ^^, £^@- xa( iOxL 
TO (i£v B0 to vna xmv A, BV' as^LexsTai jisv yag 
vnti Tav HB, BT, iei) dh 17 BH ry A' 1:0 S\ BK 
ro vitb rmv A, B^' hbqUx^tki, filv yaq vnh Tav 
HB, BJ, i:ari Si 4i BH ti; A. ro de ^A 10 v-xq xav 

10 A, /3E- teri yap ij AK, tovrdexiv rj BH, x^ A. xal 
irt bftoias 10 E& to vno tiSv A, EF' t6 ofpB vso 
rtoi' A, BF [0OV iarl ta xevao A, BA xul rp vno 
A, AE xttl hi TKF v%o A, EF. 

^Eav Kpc rafft dvo iv&eiai,, T[iij&y dh ^ iTEQa m 

15 Ttov eIs oaaStiitoTovv xfi^fiaxa, xo %E(/itx6^£vov opd^o- 
ydviov VTio Tmv dvo ev&tiwv fffov iarl Toig 
T% «TfWjTow x«l fiMtiJioi; rtov r^Ji^KTfijv neQuxofidvoie 
oQQoyaviotg' ojuq iSsi SEi%at. 



1 



20 'E«r £w#fr« yQaiifirj Tftjjfrg, tos ixvxe: 
vnb T^g oAije xRt ixardpov xav TfiijfiaToi' 
fXOftfvov opOoyoivtov fOov ^ffTi tki «jto T^g 
o/ijS riTpaytovoj. 

Ev9£ia yitQ ij AB Trrftjjff^o, (og Ixvxsv, xata xlt. 

25 r" tfijfifrof A^yra, ort ro vno xwv AB, BT ntgtix^' 

1. B2j corr. ei ZB Vin.2. 4. dK] K^ B. 6. ^AIA 
A e corr. ni.2 P. S. tol (alL) jn m. V (snpra to m. reo.>- 
7. HB] BH p. 8. t6] t» PV. 9. Post A fas. paullo 

maior linea F. td] (alt.) tto PV. 10. BH] in ras. in.2 V, 
11. To] (alt.) iw PV. 12. 'laUv ¥. t^ le orfo] lofs "nd 
it F; Tm corr. ei lOie m. 2 et poat itco rftB. V; t^ m uko nar 



»6 

J 

TS.H 



1 

;1 




ELEMENTORUM LIBER H. 121 

ducatur enim a JB ad rectam BF perpendicularis 
BZ [1, 11], et ponatur BH^ Ay et per Hrectae BF 
parallela ducatur H® [1, 31], per puncta autem z/, E^ 
r rectae BH parallelae ducantur ^K^ EJ, FG [id.]. 

A itaque JB© = JBjST + z/^ + £©. et 

^ ^ ^ ^ BG ^AxBT] nam rectis HB, BF 

comprehenditur, et BH ^ A. sed 

Bk = AxBA] nam rectis HJ5, 

K A G jB^ comprehenditur, et J5ff=^. et 
AA^AX ^E] nam dK = BH [I, 34] = A. et 
praeterea simiKter E@ = Ax EF. itaque 

A X JBr= AXB^ + AXAE + AxEr. 
Ergo si sunt duae rectae^ et altera earum in quot- 
libet partes secatur, rectangulum duabus rectis com- 
prehensum aequale est rectangulis recta non secta et 
singuUs partibus comprehensis; quod erat demonstran- 
dum. 

n. 

Si recta linea utcumque secatur, rectangulum com- 
prehensum tota et utraque parte aequale est quadrato 
totius.^) 

nam recta AB utcumque secetur in puncto F. dico, 
esse ABxBr-i- BAxAr=AB\ 



1) Arithmetice: si b-\-c =a a^ erit ab -^- ac ^=2 a*. 

p. Tw] om. F, m. 2 V. t;»o] vno tmv p. 13. tco] m. 2 
V, Toig F. vwo] vno rmv p. EFI EF nsQisroaivoig oq- 
^oyaivioig FY. y^. tco ts vno A, BJ xal tco vTro ^, JE 
Tial iti TflS vwo A^ Er F mg. m. 1. 14. ciaiv P. 16. Tofs] 
Tc3 P. vno Tf] V- in rae. p; ts vno F. 17. nsQisxofiivm 
OQd-oyavCm P. 20. ^tv;^^ V p. to] P, F m. 1, V m. 1; Ta 

Bp, F m. 2, y m. 2. 21. nsQisxofisvov oQd^OYcaviov taov] P, 
F m. 1, Vm. 1; nsQisxofisva oQd^oymvia taa Bp, PVm. 2; in F 
'Ov ter eras. 24. hv^s Vp. 



I 



122 LTorxEias f. 

(levov o^oyaviov (tera zov vitb BA, AF TtsQie^o-^. 
^ivov OQ&oytoviov tOov IiStI ra kwo tijg AB zstqiS' j 
ytSva, 

'AvttyiyQttfp&a yaQ aitb r^g AB TStQaytavov tb J 

5 AJEB, xal jjx&a Sia tov F onoziQtt zav AJ, BE ] 

jrapff^Aijiog jj FZ. 

"laov Sri iexL xo AE tofs AZ, FE. xai isti zo [ilv 
AE ro UTtb zijg AB rezQKyavov, zb Si AZ zb vnb 
tmv BA, AP aeQiBxofisvQv OQ&oyiaviov ntQiiji^stai 
10 fiiv yap iiTtb zav ^dA, AF, tStj di ^ A^ T^ AB' zb 
de FE zb vnb zav AB, BT- tati yuQ ij BE xfi AB. 
xb aga itzo tav BA, AF ^ata tov intb Trov AB, BT 
iaov iazl za aitb zijg AB zszQttyava, 

'Eav «Qa ev&eta yQafifiij r[iii&ij, mg itv^ev, 

16 r^g S^Jjs 5(«1 ixarigov zav rftijfiKttai' TteQiexofievoV' 

609oyaviov ieov iorl ra aitb t^g o^ijg tetQayavat^ 

onBQ eSsi Set^ai. 

y'- 

'Eav sv&slu yQttft(Lr{ TfirjQ-^, rog Irvxev, x& 

80 vjtb tiJB oAijg xal ivbg zav zfn}ftdtav iteQf 

exofievov 6Q9oyaviov taov ietl ta te vnlt 

zmv znTjiiarav itsQLexoftiva oQ&oyaviq} xal t^ 

aitb row jtQoetQTijfiivov tfn^jiarog zerQaycova, 

EvQeta yiiQ rj AB ret(L^a&at, mg itvxev, xaza zb 

26 F' Xiya, ort tb vno rtSv AB, BF n^Eptfxof'^'"'^ oq- 

■9'0)'iaz'£0i' taov iorl ta rs vnb zav AF, PB stBQt- 

sxoitivK» oQ&oyavia (isza zov aTtb zijg BF rszQayavov. 



L, 19. Eatociue is, 
p. SS6, 6. DoetiDS p. 386, 9. 

7. Mt] om. BPV. TE] e corr. V. eori] 



J 




ELEMENTORUM LIBER U. 123 

constraator enim in ^jB quadratum AzlEB [I, 
46]; et ducatur per F utrique A^j BE parallella 

rz [1,31]. 

itaque AE = AZ]+ FE. et AE = AB\ et 
jg AZ = BAX AF] 

nam comprehenditur rectis ^A, AF^ et 
AJ^ AB [I de£ 23]. praeterea 

rE = ABxBr\ 

^ nmLBE = AB. itaque 
BAxAr+ABxBr= AB\ 

Ergo si recta linea utcumque secatur, rectangulum 
tota et utraque parte comprehensum aequale est qua- 
drato totius; quod erat demonstrandum. 

m. 

Si recta linea utcumque secatur, rectangulum tota 
et alterutra parte comprehensum aequale est rectan- 
gulo partibus comprehenso et quadrato partis nomi- 
natae.^) 

recta enim AB utcumque secetur in puneto JT. 
dico, esse ^J5 X J5r= AFxTB + BFK 



1) Arithmetice : (a + 6) a « a& + ^*- 



8. AZ] dno tfjg AZ F, 10. AJ] JA F. ^ 13. iat^v P. 

14. yQa fiftri] del. in P. itvxs Vp. to] ta Bp, F m. 2, V 
m. 2. 15. nsQisx6(t8va og^oymvia tca Bp, F m. 2, Y m. 2. 
19. itvxs Vp. 21. htCv P. xs] supra m. rec. F. 23. 
cino] corr. ex vno p. nQOSi^rniivov] ngo- m. 2 V. 24. 

^tvxs Vp. 25. r srjiisiov Vp. 26. ts] om. Pp. AF] 

r in ras. V. nsQLsxo^ivm oQO^oycovia}] om. Bp. 



124 STOIXEIiiN |S'. V 

'AvayEYQu^&ia yap aito xiig FB nTQiiyavov to 
rJEB, xttl di^x^a 7j E^ inl lo Z, xal Sia. zov A 
oiroTspK xav rtni, BE jraQakXiilos VX&Ci} ^ AZ. l'aov 
6tJ iett to AE tols AA, TE' xat iCTt rb (liv AE 
5 To vjih zmv AB, BF xsQifxoii^vov OQ&oymviov tccqi- 
ixBtai (liv yecQ vnb tmv AB, BE, tSrj Si i) BE Tij 
BT- To dl AA to vnb tdv AF, FB- rat] ya^ rj 
j^r tjj FB' to Se AB to a%b i^g FS TCZQaymvoV 
To tep« vxo tiav AB, BP ^£qie%o^evov OQ&oyeovtov 

10 iflov iotl rp vitb riov AT, FB 3tcQi£xfi{iEva oQ^oym*^^ 
via [leta tot) ajtb t^; BF TETQaycovov. | 

'Eav aga tv&tia y^aiiii'^ t(irj&?i, cSg itvxev, rh 
vab r^s oltjg xal ivbg t(5v tfiijjiaTtav asifiixo^Evav 
oQ&oyfovtov t&ov ictl ta xs vitb tav TfitjfiaTmv jieqi- 

15 Exoiiiva OQ&oyavCa xal rra anb toi; nQOHQ^ri^ivov | 
tft^futtos TErpKyoJvp" oreep ^Sei SEt^ai. 



I 



'Eav EV&Eta yQa[i(ii} XfiTi&^, rog etvxev, to 
«JTO r^s oAijs tftpayrawo» fffov ieti rofg t£ 
20 «reo Ttov x(iiifiaTav TSTQaytovotg xal t0 Slg 
vKo Tmv t(iTj[iatav «tQtsxOfiivip oQ&oyavia. 

Ev&iia yap yptt(i(trj ^ AB TJt/ijJo^w , rog hvxEV, 
XKTtt To r. Xiya, oti tb ditb tfjs '^B reTQaymvav 
teov iatl tolg t£ airo tav AF, FB tEZQttyiavotg xal 
25 ip 6ls v%b tmv AP, PB steptExoftrVa OQ&oyavia. 

'AvayEy^aip&io yag anb xijs AB tEtffdyavov t&b 



IT. Theon in Ptolem. p. 184. BoetiuB p. i 



. r^jBE B, m. a V. 
itt m. 2 F. 8. FB] 




ELEMENTORUM LIBER n. 125 

construatur enim in FB quadratum FjdEB [I, 46], 
et educatur E/J ad Z, et per^ utrique Fjd, BE par- 
allela ducatur AZ [1, 31]. itaque AE — ^z/ + FE. 
p -g et AE^ ABxBF] nam comprehen- 
ditur rectis AB, BE, et JBJB — BR 
et ji^ — jirxTB] nam jr^TB. 
et jdB = rBK itaque 

^B X Br— ^rx FB + BFK 
Ergo si recta linea utcumque secatur, rectangulum 
tota et alterutra parte comprehensum aequale est rect- 
angulo partibus comprehenso et quadrato partis no- 
minatae; quod erat demonstrandum. 

IV. 

Si recta linea utcumque secatur, quadratum totius 
aequale est quadratis partium et duplo rectangulo par- 
tibus comprehenso.*) 

nam recta linea AB secetur utcumque in F. dico, 
esse AB^ = AF^ + TB^ + 2 AFx FB, 

construatur enim in AB quadratum A^dEB [1,46], 



1) (a + 6)» = a> + 6« + 2a6. 



BT F. rJB] e corr. p. 11. JBT] FB Pp; corr. ex AT F 
m. 2. 12. hvxBv] PF, B sed v eras.; ixvxB Vp. 13. vno] 
V' e corr. p. 15. nqoHf^rnLBvov] nqo- in.2 V. 18. ixvxB 
Vp, B e corr. 22. yapl m. 2 F. hv%B Vp, B e corr. ^ 

23. r arifiBtov V. 24. iatLv P. rfi] om. V. TeT^ayco- 
voig — 25. FB] mg. m. 1 P. 25. tcov] om. P. 



126 STOEXEIiiN p'. ■ 

A^EB, xal tat^Evx&a ■^ B^, xal Sia [itv tou T 
o^OTdpa Tmv Azi, EB Kap«AAjjAog ^x&a rj FZ, Stit 
di row H oKor^pa tav AB, ^E xa^akktilos ^fro 17 
®K, xal Ind ^a^dlXtiKos ioriv tj FZ rfj AJ, xal 
b tlg avzas iiiTitazmxtv ^ BjJ, ^ ixios yavLK i ixo 
FHB iaij iarl t^ ivzds xal aJiEvavziov t^ VTto A^B, 
aXK' ij vjto A/IB tfi vno AB^ ieriv fstj, ixtl ;(«l 
aXevffa j] BA TJJ A^ idtiv taTj' xrI 7; vxo FHB 
«Qa yavia Ty ioto HBV ieziv i'arj- &azB xal nXiVQa 

10 ij BF TtXsvga r^ FH iativ (ati' alk' 17 \iev FB z^ M 
HK iaziv i'arj, f} Si FH z^ KB' xal gj HK apa t^ I 
KB iariv ieti' {aoitlBVQOv a^a iazl z6 rHKB. Xiyea " 
6^, ozt xal OQ&oymviov. i%El j-ap xaQixXlriXos iaztv 
7j FH Tf) 5^ [xal eIs avzas i^nintaxBv tv^Bta ^ 

15 rB\, af «p« uiro KBF, HFB ymvica dvo opdnfs 
tiaiv teai. oQ&Tj di 7] vKo KBT' op&Tj Kp« xat 7/ 
iwo BVH- mezB xal at aTtevavziov at vao FHK, 
HKB opfl-ai Eiai.v. oQ^^oymvtov aqa iezl zo FHKB' 
iSsix^V ^^ ""^ CeonXEvQov' zerQayavov aga iezCv 

20 xai ieziv «JTO r^g FB. Sia ta avza Si} xal z6 &Z 
zezpayavov ieziv' xai ieztv aab z^g ®H, zovzietiv 
[rajro] r^g AF' ta aga @Z, KF rsTQaytova «wo rav 
jr, VB aieiv. xal insl teov iezl z6 A H z^ HE, 
xai iezi ro AH ro vxo tmv AF, FB' i^arj yap tj HF 

25 zri rS ■ xal z6 HE aga Caov iazl za vzo AT, TB- 
za a(fu AH, HE tea iazl rp dlg vito tmv AF, FB, _ 

2. rZ] ZrZ P. Sia SiJ wl e^a p. 3. AB] B ia ^ 

raB. p. Post jineoUiilos in P est yqafiiiov punotia delet. 

4. rZ] corr. ei ZT F. 5. BJ] JB p. 7. dUo Vp. 

10, alla PVp. 11. KB] B e corr. p;BK P. 12. 

iotir ro.)] om. p. ieri] taziv P. 13. Sn^ om. F. 1*. 



ELEMENTORUM LBBER IL 127 

et ducatur J5^, et per rutrique AA^ EB parallela du- 
catur rZ [I, 30et31], per H autem utrique AB, ^E 
parallela ducatur GK. et quoniam JTZ rectae A^ 
parallela est, et in eas incidit B^, angulus exterior 
FHB aequalis est angulo interiori et opposito A^B 
[1,29]. uerum L A/IB = AB^, quoniam BA = AA 
[1,5]. quare etiam L THB = HBF. itaque etiam 

J5r=rff [1,6]. sed etiam FB^HK 
4 r g [1, 34] et FH = KB [id.]. quare etiam 

jf; HK = jSl JB. itaque aequilaterum est 

FHKB, dico, idem rectangulum esse. 

nam quoniam FH rectae BK paral- 
■g~jfe lela est, erunt KBF -{' HFB duobus 

rectis aequales [I, 29]. uerum L KBF 
rectus esi itaque etiam L BFH rectus. quare etiam 
oppositi anguli FHK, HKB recti sunt [I, 34]. ergo 
FHKB rectangulum est. sed demonstratum est, idem 
aequilaterum esse. ergo quadratum est; et in FB con- 
structum est. eadem de causa etiam ®Z quadratum 
est; et in &H, hoc est AF [I, 34] constructum est. 
itaque quadrata 0Z, KF in -^r, FB constructa sunt. 
et quoniam AH= HE [1,43], et AH=ArxrB 




Tictletg avxas iiinint(o%sv sv&sia rj FB] add.Theon? (BF^^p); 
mg. m. 2 P. i(tninT<o%£v'] euan. F; fvinsasv B. svd^sia] 
om. BF. 16. FB] B eras. p. HTB] BVH P. Svo\ 

dvclv Vp. 16. fffat slciv Vp. 17. af] (prius) om. F. 

18. iffrt] icxCv P. 19. icxCv] PF; Iffrt uulgo. 20. rJB] 

corr. ex Br m. 2 V; Br p. (9Z] e corr. p. 21. ^cJTiv] 

(prius) PF; iisxi uulgo. ©H] He F. 22. ano\ om. P; 

in F eras. KV] FK Pp. 23. sCatv] F; iaxiv P; slai 

uulgo. iaxQ iaxCv P. 24. sativ P. Ante HF ras. 1 

litt. F. 26. Post apa ras. V. iaxCv ^Y. AV] t^v AF 
Vp, F m. 2. 26. AH] corr. ex i4JB p. iaxCv P. 



128 STODCEmN f. ■ 

itStt Si Kal Ta @Z, PK zsTQuyava a%o xav AF^FB' 

xa UQa Tieeaga tk &Z, FK, AH, HE i^aa iorl rof? 

Tt djio Tav AP, rB TirQaymvoig xal rra dtg vjio 

Tcov AF, FB niQU%o^iva OQ&oyavia. aXi.a r« ©Z, 

5 rK, AH, HE o?^ov cffii ro A^EB, S ietiv «wo 

T^S -^B TiTQayavov to «pa ano rijg AB Tstpayavov 

t€ov ierl Totg i^£ «Jto Tmv AV, FB TBrpaycovoiq xal 

Tt5 Slg vTto tav AF, FB mpt,exo{iiva OQ&oyavia, 

'Eav aqa ev&sta yQR^iJ,rj z^tjd-^, tas irvxev, to aao 

10 r^s oAijg t£Tpayavov teov iozl rotg tt dzo xav t(1)i- 

(latav tetffaytovotg xal ta Slg vxb rav T^rjftttTiav 

jTEptf^dop^vra oQ&oymvia' OTtsQ tSBt Ssi^ai. 

IHo QiQfia. 
^Ex Sij tovtov ipavEQov, oTi iv totg zetQayavott 
15 jjrop/otg ta ntQl tijv Sid[tEtQov naQttllTjXoyQafifia te- 
tQuyavd iettv']. 



i 



'Edv Bv%ila yQanfiii r^Tj&ij elg laa xal^ 
avtStt, To v%b rav dvCoav T^g oXtjq t(if}itdtav 
asQi.ex^f-^''"^^ oQ&oymviov ft£ra ToiJ dno tijs 
fieta^ii Tc5v TOjimv ttrQttyavov teov icrl ra 
dico rijg ijfitaeitts retQttymva. ■ 

Evd-eta yuQ Tig rj AB TEr(i^a9-a tls fitv lea xffir« 



IV. «dp. De Proclo p, 304 a. ad IV, 15., V. Boetijri 



1. lottv P. TO] TD Fr corr. m. 2. Tttgdyavov F; 

corr, m. 2. 2. ib] (alt.) om. F. iazh P. 3. «] m. 2 

V. 4. op&oyojna q3. tu] to TtairojB P. 0Z] in 

raa. V; Z0 B, 6, HE] H e cott. p. iatit P. AJEE 



ELEMENTORUM LIBER II. ^ 129 

(nam HF = rs), erit etiam HE^AFx FB, ita- 
que ^lf + ^^ = 2 AFx FB. uerum etiam qua- 
drata 0Z^ VK m AF, FB constructa snnt ergo 

»z + rK+AH+HE^An+rB^+2ArxrB. 

sed »Z + rJC + AH+HE — AAEB — AB\ ita- 
qne ^5« — AI^ + rS^ + 2 AFx FB. 

Ergo ai recta linea utcunque secatur, quadratum 
totius aequale est quadratis partium et duplo rectan- 
gulo partibus comprehenso; quod erat demonstran^ 
dum.^) 

y. 

8i recta linea in partes aequales et inaequales se- 
catur, rectangulum inaequalibus partibus totius com- 
prebensum cum quadrato rectae inter sectiones positae 
aequale est quadrato dimidiae. ') 

nam recta quaelibet AB ia aequales partes sece- 



1) Etiun Gampanas hic duas demonstrationeB habet, qua- 
ram prior reiectae, altera neque huic neque reiectae siinilis 
est. de hac habet: ,,sed hac uia non patet correlarium, sicut 
uia praecedenti patet, unde prima est autori ma^ consona." 
nam corollarium et ipse habet. itaque fortasse Theone anti- 
quius est. 



^.»'+(4^-')'-m"- 



TstQctYmvov V. Q, AB tstQdyatvov] (prius) mg. m. 2 V; in 

textu ras. 2 — 3 litt. tstgdyoDvov] mg. m. 2 F. 7. iativ P. 

T«l om. p. tmv] m. 2 F. 9. itvxev'] B; hvxs uulgo. 
10. iat£v P. te] om. p. 12. Sequitur alia demonst^tio, 
quam Augustum secutus in appendicem reieci. 18. noffiafia 
— 16. iativ] add. Theon? (BFVp); mg. m. rec. P. 14. tov- 
tmv P. (pavsQOv iativ V. 18. sts] supra m. 1 V. 19. 

tlg avusa p. 21. iatCv P. 

Euolides, edd. Heiberg et Menge. 9 



130 STOIXEIflN p'. ' 

To -T, sis ^i Kviea xutk to ^' kdyco, ott lo vno rav 
Ad^ ^R asQtaxoii^vov og&oymviov fieTu zov «mo t^g 
rzJ rETQaytovov Coov ioxl xm aito r^g FB TfTpayajVto. 
'jivayty^Kfp^a yap emo T^g FB tiTpaycavov to 
6 FEZB, xal iatttvx&ixt ^ BE, xa\ dia iihv roii ^ 
onotd^a tmv FE, BZ aapaXlTjlog ^j;3^ra ]j ^H, Sia 
Si Toii & bnoztQa twv AB, EZ naQaXlriko^ xaXiv 
•lYfpfD i\ KM, xa\ adltv Sia tow j4 onotiQa ttov 
FA, BM xaQaXktjlog ijx^ea r} AK. xal iTitl Haov 

10 ietl zb r® xagtt7cl^(/0}iia tm ®Z naqaTtKi^Qmfiati, 
xoivbv TtQoextie&a lo z/M' o^oi' Kpce tb FM oka 
ta ^Z teov iatlv. aXXa to VM tm AA ieov istiv, 
iaal xal tj AV ty FB istiv tetj' xal tb AA ap« Tt5 
AZ iHov iariv. xoivbv nQoGxeie&a to r&' o^ov «po! 

16 TO A@ ta MNS yvia(iovt t«ov iativ. aXXa tb A& 
To vnco tmv AJ, /IB iotiv fffjj yag rj A® tfj /JB' 
xttX 6 MN3 aptt yvtafKov fffog iatl tS vnb A^,^B, 
xoivbv TCQoaxtia&io ro AH, o itstiv teov r^ anb r^g 
r^' o aQtt MNS yvaniav xal tb AH iaa iatl tp 

30 vico t£v A^, /JB fctQiBxoiiiva opftoyojvtp xal tm 
«JTO T^s r^ zttQttycovc}. dXXa b MNS yvdfiaiv xal 
TO AH oAov iati tb FEZB TEtpdyavov, o iattv djco 
tijs rB' To aqa vnb tavAJ,^B %BQUx6p>Bvov oq- 
^oyaviov fieta tov dxb t^? Fz/ TfTpaytovov taov iazl 

25 T^ dnb tr^s fB titQttyfovc}. 



3. iexlv F. ztxttttiivcp] om. B; comp. add. m. 2 P. 

B. rEZS] in ras. p. BE] B in ras, F. 6. BZ] ZB P. 
aUc Si] ROil Sia V. 7, JwJlt*] om. p, m. 2 V. 8. kbI jrrili* 
— 9. l AK^ mg. m. leo. P. 10. eZ] Z© F. 12, teov Ux{v\ 
(alt.) Lziv fflov V. 13. inU - rar;] mg. m. 2 V (rai) htf). 
14, Uxiv fooy ¥. iazCv'] P, comp. m. 2 F; hxi Bp. 16. 



i 



J 



ELEMENTORUM LIBER U. 



131 



Fi I^ in luaequaieB autem in iJ. dico, esse 
^^x^B-\- r^' = rs-. 
instruatur eniiu iu FB quadratum FEZB [I, 46], 
et ducatur BE, et per zl utrique FE, BZ paraJlela 
ducatur JH, per 8 autem utrique ^B, EZ parallela 
ducatur KM [T, 30. 31], et rursus per^ utrique rA,BM 
parallela ducatur AK. et quoniam F® = 8Z [1,43], 
Gommune adiiciatui ^Af. itaque VM = idZ. uerum 
„ . rM= AA, quoniam 



1 


a'/ 


r ^ 


r^ 



^Ffi. quare etiam 

AA = .i/Z. commune 

adiiciatur F®. itaqne 

-Uf vi ®=MiV& gnomoni.^) 

uenim 

^ ^ ' ^@ = ^^X^JJ 

I (nam ^&^^JB); quare etiam MNS = AJx.dB. 



I Gommune adiiciatur y/if, quod aequale est F^^. ita- 
que MNS + ^H= v^^ X JB + T^*. aed 

Jtf JVS + ^H = FEZB =- rB\ 
itaque ^^ X ^B + rzT' = TB». 

1) CDtn litteia M iu figtira, quam ex ed. Baail. recepimns, 
bis Qsarpetur, uon aine causa pro MNS a Gregorio scriptnm 
est NSO, nt prop. VI, sed uon audeo contra codd, mutare. 



MNS yviittovi] P; CumpanuB; -JZ KolJ^ Theon {BFY; pro 
J,l in F JA; JA xal JZ p), lo .<»] lo ui* AB Bp. 

16. yae ^] ^ y<ip P- .J^] J6 p. J£]'^e iati p. 
Post JB add. Theon: xa Se Z J, J^ hziv o MNS yymiumv 
B CZ.J-1), F. V (priue J in ras.), p (i M2VS hu); om. P. 

17. M«r| om. p. lo] rd F. oso iiS* p. 19. ^ot(V P. 
80. xtifitxa^ivta» QifioyiBvlmv F, 21. all.«] all' F; aUd 

■ «k/ V. 23. rBl post raa. 1 litt. V; BT p. 2*. oni t^f] 

■ BUpra m. 2 F; ani P. ini^i' PV. 



1 



1S2 STOIXEIiiN |J'. 

'Eav «Qa tv&eta yQafifiii r^i?#f; sig tea xal aviaa, 
10 vno Tiav avCew r^^ oiijg vp,7iiidTiov jtepiEj^d/iEvow 
dpdoyiowiov fiBta rov «rco r^s fiir«|i' rtov roftrai' re- 
rpoywfoii i'ffoi' ^flrl r^ awo r^g Tjfiieeias TBrgaymv^. 

s'. 
'£av cij#f(a ygaftiii] rfiTi9rj Siia, xq^O0ts& 
ds rts avT^ Ev^^sta i%' sv9sias, to vito rijs 
Siije ffvv r^ x(foaxei,[iivTi x«l r^g OTpooxEj 

10 xspisxofisvov op^oyojvtoj' ftfra rou «ji 
rilii0sias veTQaymvov taov sGxX tm an 
avyxsiiiivjfg ix re r^g fi(ii.0siae xal tijg npoff-, 
xetfidvtjs tszQttyavc}. 

Ev^sta ydg rtg ^ jiS TCrfMjffflra ff();a KOTW ro i 

IB atjftstov, nQoexsiU^m di ttg avtji sv&sta iit sv^eia^ ' 
i\ SA' Xsyfo, otL ro vxo tmv A^, ^B 7tsQisx6(isvov 
6p&oytavtov (ttrB rou anb r^s FS tstpaydvov Itfov 
iatl rt5 anb tijg fz/ tstpaytova. 

'JvaysyQaip&m yuQ airo t^g r^ tttQdyatvov. iA'l 

ao VEZJ, xal ins^svx^o} tj ^B, xal Sta ftsv tov B"^ 
ejjfieiov hnotsQa tmv EP, ^Z zaQalXrjlos tjx&^ ^ 
BH, Sia dh tov ® arjiisiov baotsQa tmv AB, EZ 
itttQwlltjieg ^x^^ V -K^) "'f^ ^iT' ^'« row A onotiQa 
tav FA, zlM aaQalXrjXog %*(a 7] AK. 

26 'Eitsl ovv ta^rj iatlv fi AF r^ FS, Caov iarl xal 
tb AA ta r@. dXXa ro F® zm ®Z feov ietiv. xmi 



L 



VI. Schol. in Archim. III p. 383. BoetiuH p. 385,22. 

1. 7liau4 P- . tts avtaa p. 4. ieziv PT. 8 

tv9eiiti, M vito] in raa. V, 9. «pooK ftfii*»] -e- Bupra 
j[po)irifi*'v)]S V, et p fled corr. m. 1. 11, l^iv V. 

npaiMEifiEvi;;] -a- inBert. p. Post hoc nerhum legitur loi 



1 
I 

VS M 

i 

iag ■ 

V 
V 

I 



I 



I 



ELEMENTORUM IJBER U. 133 

Srgo si recta linea in partes aequales et inae- 
quales secatur, rectangulum partibus inaequalibus to- 
tius eompreheusiim cum qoadrato reetae inter secti' 
ones positae aequale est quadrato dimidiae; quod erat 
demo u str andum . 

VI. 

Si recta linea in duas partea aequales secatur, et 
ajia quaedam recta et in directum adiicitur, rectaii- 
galum tota cum adiecta et adiecta comprehensum cum 
quadrato dimidiae aequale est quadrato in dimidia 
adiectaque descripto. ') 

nam recta. aliqua JB 



1 



^'^ partes aequales secetur in puncto 
J', et alia quaedam recta B^ ei 
in directum adiiciatur, dico, ease 

construatur enim in FJ quadratum FEZJ, et du- 
catur iJE, et per B punctum utrique EF, jJZ par- 
allela ducatur BH, per autem puuctum utriqae 
AB, EZ parallela ducatur KM, et praeterea per A 
utrique r//, jJM parallela ducatur j4K. iam quoniam 
Ar=^rB, erit etiam AA = r&. sed F® = ®Z [I, 
43]. quare otiam AA= &Z. commune adiiciatur FM. 

I 1) (8a-|-6)6 + a»-{a-|-6)'. 

fuas avaYfuipivxi in p, P mg'. m. rec, Zambettoi om. fioetius, 
CampanuE, P m. 1, fi, V m. 1; in F fnit a m. 1 (rest&nt. . 
Uffcetpfvn), eei TtToaytavia 7; log ttno fuog V mg. m. 2. 
18. i<it(v V. 20. liciifvx»o — 21. JZ] mg. m. rec. P. 
21. EF] FE Pp. JZ] ZiJ (p. a-2. ffTjiit^ov] om. p. 

AB} jIBJ p, AJ P. 2b. AT} in ras. V. lexiv V. 

26. bUo] alXa xai F. firov faiiv] P; taov F, iW iaxi B; 

imiv teov Vp. 



t 



134 STOIXEiaN ^'. 

zo AA aQBc za &Z isxiv i6ov. xoivov Jtpoexeiiti^a 
ro FM- oAov KQce ro j4M rra N30 yvdfiovi ioxiv 
tCov. ttXXa To j4 M icri zb vito tav A/J. /iB' fffTj 
yaff isztv ^ iJM t^ ^B' xal 6 NSO aqa yvmiiiav 
6 rSos isrl TQ vzo xmv A^, ^B \ni^iEyO]iiv(p 6q90' 
yavia]. xoivbv nQoSxtis&ei ro AH, o istiv tsov rp 
aito tijs Br terpayrawp ■ ro apw iwo roiw ^^, /IB 
xeifiexofitvov o^&oyaviov (tera tov aao t^s ^^ ^*- 
rqayiDvov iaov istl ra NSO yvmfiovt. xal ta AH. 

10 aiika 6 NSO yvwiitov xal r< AH oAoi' istl t6 PEZA 
ttTQuymvov, o istiv aao r^g F^' t6 aQa vito tmv 
Ajd, ^B mffiexofisvov 6p&oy(6viov ftsxa rou ajro 
T^S rs tetQaymvov Csov istl ta kjto rijs FA tttQtt- 
ymvo}. 
5 'Eav af/a sv&sta yQa(L(i.i] Tftij^S^ ^^X^y «Qoste^ 
di zig avtf) ev&ela in ev&eias, t6 vao t^g oltjg Svv 
T]i itposxsiftivij xal t^g XQoSxeijisvTii nsQisxoftsvov 
OQ&oymviov ftsta T0J3 dno r^s ijiitSeias tetQuymvov 
iSov istl Tco ttJto t^s svyxeifiivris ix te r^s rifiissias 

20 xtd tijs JtQoSxeifisvtig tszQayavm " oTteQ iSet dst^ttt,. 



'Eav ev^^ela yQan[ii] tftrj&f), tog itvxev, to 
axo Trjs oAijs xal t6 dtp' ivos tmv Tnij(i<irmv 
ta SvvaitipOTsgtt rstQaytova isa istl ra te Sls 
Wih vith Tij; oAijs xa\ tou eiQrjltivov rft^ftatos Jtsffi- 
tXO^ivia OQ^aymviip xaX rp ««6 tou Kotaov 
TfiTJfitEToff terQttymva>. 

EvQsitt yd(f tte ij AB teTj^^ffOo), mg h:v%sv, xatk 



I 



. AA] AA P. apo] om, F. »Z] corr. mt Ze V. 



ELEBiENTORUM LIBER H. 135 

itaque AM^ NSO. uerum AM^ AJ X AE\ nam 
JM^JB, quare eidam NSO = A/1 X /1B. com- 
mune adiiciatur AHy quod est BF^, itaque 
AJ>:JB + FB^ — NSO + AH. 
sed NSO + AH^ FEZA = F/fi. erit igitur 

^^ X z/5 + FB^ = r^. 
Ergo si recta linea in duas partes aequales seca- 
tur^ et alia quaedam recta ei in directum adiicitur^ 
rectangulum tota cum adiecta et adiecta comprehen- 
sum cum quadrato dimidiae aequale est quadrato in 
dimidia adiectaque descripto; quod erat demonstran- 
dum. 

VIL 

Si recta linea utcunque secatur^ quadratum totius 
et quadratum alterutrius partis simul sumpta aequalia 
sunt duplo rectangulo tota et parte nominata com- 
prehenso cum quadrato reliquae partis.^) 



1) (a+ 6)« + a« =- 2 (a + 6) a + &*. 



2. TM] in ras. V. iNT^O] N in ras. V. yvdiioDvi F. 

3. iariv FV. 4.. JB] B eras. V. NSIO] N corr. ex M Y 
6. iativ V. TtBQisxofisvq) doO^oyfloWco] om. Pp. 8. FB] 

BF Y. xstQuymvan tp. 9. sativ F V. 10. iativ V. 
rEZ^] Z in ras. V. 11. FJ] in ras. V. 12. OQd-oym- 

vLov] 0^0- in ras. m. 1 p. 18. FB] BF Vp. iativ V. 
dno t^s rj] FB (p seq. lacuna. 15. yQccfifiT^] seq. ras. 4 

litt V. nQoad^ij P. 17. nQoa%sifisvjj] a inserC. m. 1 p, ut 
breui post et lin. 20. 19. iatCv V. 20. Ante tstQccymvot 

in Fp: mg ano fiLCcg dvccyQa(psvtii idem post tstQccyoivai in- 
sert. in V m. 1? onsQ idsi Sst^at] :— BF; om. V. ' 22. 
itvxs p. 24. iat^v F. ts] ds P; corr. m. 1. 28. itvxs 
Fp. 



136 STOIXEUiN p\ H 

ro r etifittov liyfa, oti tcc koto t<dv AB, BF tezfa- 
ycova i'6a iatl ta ts Slg ^wo tcov AB^ BF it£Qie%o- 
(livci op&oyiovic} xal t<p ano t% FA tEt^aytova. 
'AvayeyQoKpQto yaQ «reo T^g AB TSTgdymvov tb 
5 AAEB- xal xaTayeyjfaip&a to Sj;^^. 

'Enel ovv teov ierl zb AH ta HE, Kotvbv mpoO- 
xdtO^O) TO rZ' oAov ttffa tb AZ oip Tp FE teov 
ietCv ta aga AZ, PE Siniaaia iott zoii AZ. aXXa 
tu AZ, FE 6 KAM iezi yvfoiitov xal tb FZ rstQa- 

10 yesvov 6 KAM apa yvm^iav xal to FZ dtJiXaeia 
ieti Toii AZ. ifftt S\ zov AZ StnXueiov xaX zb 8Xs 
VTtb Ttov AB, BF- Tffij yap ij BZ t^ BF' 6 Kpa 
KAM yvmfieav xal tb FZ tatifaycivov fffov ietl r^ 
SXq VTtb zmv AB, BV. xoiybv nQoaxeie&a z6 ^H, o 

15 iativ axb r^g A F tstQaycavov 6 cpn KAM yvdfieiv 
xal T« BH, HA teTQdyava Hea ietl ta re Slg vxb 
rav AB, BV neQisxo^ivm oQ&oyavCa xaX zm anb 
T^s," AV tttQKydvip. aXXa 6 KAM yvta^cov xaX tcc 
BH, HA tetQdymva oAov iarl ro AAEB xal t6 FZ, 

20 K ietiv nitb tmv AB, BV tiZQttyoiva- tu kqu dnb 
tiov AB, BP tiTQuycovtt tea ierl t^[t€]SXs vnb tcov 
AB, Br neQiexofiBva OQ&oymvip fieta rov uab t^g 
AT tsrQaydvov. 

'Edv UQtt ev&eia yQafifiri Tftijfr^, tog ^zvxev, to 

25 ditb zijs oAjjs xaX ro «y' ivbg teov tftijftdriav rd tJw- 
ufitpoteQU reTQaymvtt iea iozl rra te SXg vnb r^s oXijs 
xal TOti elQtjfiivov zfit^iiatos X£QiBxO(iivp oQ&oyavip 
xttl T^ ditb ToiJ XoiJtov tiitjfiKzos tezQayava * ontQ 
iSsi Stt^ai. 





ELEMENTORUM UBER U. 137 

nam recta AB secetor utcunque in puncto F. dico, 
esse AB^ + BI^ = 2ABx Br + TA^ 

construatur enim in ^^ quadratum A/iEBy et 
describatur figura.^) iam quoniam AH = HE [I, 43], 
commune adiiciatur FZ. itaque AZ = FE. quare 

AZ-\- rE=2 AZ. • uerum 

Az + rE = KAM + rz. 

^ itaque KAM + TZ = 2 AZ. sed 

2ABxBr=2AZ', nam BZ = 5r. 

itaque KAM + FZ = 2 ABX BF. 

^ JB commune adiiciatur AH^ quod est AF^. 
itaque KAM + BH + HA = 2 AB X BF + AI^. 
sed KAM + BH + HA = AAEB + FZ = AB^ 
+ BI^. erunt igitur 

AB^ + BI^ = 2ABxBr+ AFK 
Ergo si recta linea utcunque secatur, quadratum 
totius et quadratum alterutrius partis aequalia sunt 
rectangulo tota et parte nominata comprehenso cum 
quadrato reliquae partis; quod erat demonstrandum. 

1) Sc. eadem, quae in praecedentibus propositionibus, ita 
ut ducatur diametrus JB^ et per T rectis AJ^BE parallela 
rjV, per H rectis AB^ JE parallela BZ. 

I#r/ B. Tff] TO p. 6inXccciov p. icuv PY. AZ} 

corr. ex JBZ m. 1 p. 9. ra] to p et post ras. 2 Htt. F. 
icxi] iaziv V, supra m. 2 F. 10. StnXdaiov p. 11. iativ 

FV. Post icti 1 litt. erae. V. tov] e corr. p. 12. BZl 
ZB p. 13. iatLv V. tm] corr. ex to m. 2 V. 14. BF] 
Br nsQi^sxofiivco oQ^^oyaiviai p. 16. iativ FV. ts] ds P; 
corr. m.l.^ 18. aU* F. ' 19. iarivY. 20. «] supra m. 1 
F. ano] toc ano F. tmv] tijg comp. p. BF] om. P; 

corr. m. rec. 21. iat£v V {v eras). ts] om. P. 22. 

nsQisxofisva (p. fista tov] xal tm p. 23. tstQaymvoo p. 
24. hvxs p. 26. iativ V. 27/ nQost,(^fisvov P. 



ETOIXEiaN ^ . 



V ■ 
Eav sv&sta yganfiij r[iij&fj, fos Stv-(_EV, 
TBTgaitig viti} t^g Zkrjq x«t evos rav riirjfitttiov ' 
jrfptsxojtfvo" OQ&oycovi-ov (leza: tov axo rov 
) Aotsroi) TitijfiKTog tBtQaycovov fffov istl ta a%6 
TE t^g oAjjg xal tou ctgrinivov Tfi^ftarog ms 
aab (ti.dg avayQuqitvtt tsT^aysav^. ' 

EvQsia ya(f tig ^ j4B rttii-^a&co, ms hv^tv, xa- 

Tu To r ff)]ft£rov ■ Xiya, OTt to T£rp«xtg w?r6 Ttov y4B, 

10 -SJ' xSQiex^f^^^^ov OQ&^oycoviov (leta ToiJ ajTo T^g j^F 

tETQayiovov i'6ov detl ra ano Tijg AB, BF rag tJnio 

fiiag ttvayQttfpivri rttpayiova. 

'EM^£^X'rje9io yuQ iz' sv&Eiag [zij AB tv&EtcL] q ' 
B^, xttl xtCe&co ry VB /ffj; ^ B^, xal avaysyqa^&a 
5 anb zijg AA tetpdyiovov rb AEZA,xaXxaray£yi/dip&ia 
Sta^.avv ro Oxil(ia. 

'Eml ovv tari ietlv ^ rB ry BA, akXa ^ fih/ FB 
Tjj HK iettv tejj, tj Si BJ rjj KiV, xal rj HK aga t^ 
KN leTiv feij. Sta ta avra 6i} xai i) IIP rrj PO 
20 iortj' fffij. xal iael tOti ierlv i] BF r^ BA, i] Si 
HK r^ KN, Ceov aga ierl xal to (lIv 'fK rp K^, 
t6 Si HP T(5 PN. aXXa ro FK ra PN ietiv taov 
aapaaXrjQiofiara yaQ row FO naQaXlr}loygii(i(iov xal 
tb KA ftpa T(5 HP leov iarCv ' to; tiaeaQa uQa ra 
I S6 AK, TK, HP. PN i'ea aXX^^Xoig ietCv. rit Ti 



postea odd. . 



3. TETpaxiie V, corr. m. S. 5. iarCv FV. 
aito Pp; uKo F. T. Kva^^ea^fWi] -u 

8, {Toxt p. S. Hipfinije V; corr. m. 3. 
i«i(DvmvQ> p. iativ V. 13. yap] oin. F. t^ AB 

tv9cCa\ Theon? (BFVp; cv9eCa B); tn. rec. P. 14. /'oi; ig 

rs P. FB] Br F. SJ]'JB V; corr. m. 2. 17. rBJ 
Br P. bU' P. 18. BJ] JB V, coiT. m. 2. KN] 



ELEHENTORUH LIBER n. 139 

vm. 

Si recta linea utcunque secatur, quadmplum rect- 
angalom tota et alterutra parte comprelieDBum cum 
quadrato reliqnae partiB aeqnale eat quadrato in tota 
aimul cum parte nominata constructo. ') 

nam recta AB atcunque secetur in pnncto F. dico, 
esae i ^BxBr+ AV* = {AB + BV)*. 

producatnr enim in directum AB, ut &at Bid, et 
ponatur £z/ — VB, et in A^d construatur quadratudi 
AEZ^, et fignra dnplex describatur.*} 

iam qaoniam FB -^ B^, et 
FB = HK, BA~ KN, erit etiam 
^■, y HK — KN: eadem de cauBa etiam 
7IP= PO. etquoniamBr— B^, 
' HK =- KN, erit FK — KJ, 
HP = PN. uerum FK = PN; 
nam supplementa sunt parallelo- 
grammi FO [I, 43]. qaare etiam 
K^ = HP. ergo quattuor ^K, TK, HP, PN 

VIII. Pappna V p. 428, 21. 



r — 



1) 4(a + 6)a + 6'-[(<. + b)+a]«. 

9) H. e. dDcta diametTO JE, dncantnr BA, r9 rectis JZ, 
AE porallelae, MN et SO rectis AA, £Z; a. p. IST not. 1; 
sed ibi dnae tkntutn paiallelae dacnntur, bic qnattaor; qaaie 
fignra daplex noc&tar. 

JtH V, conr. m. 8. HK]ecorr.V. «ea] PFp; oin.flV. 19. 
KN]KHY; corr. m. 2. «oi ij HP] in ras. V. 20. ij] ij iiiv 
Bp. Sr] rs p. 21. iaziy PFV. ■=/] om. B. fiiv] 
om. P. JCJ] B^ P; in raa. egt io V. 22. PN] (priaB) NP Pp. 
I>ein add. firov in nu. V. 2S. ya^ tlm p. 24. ta] corr. ei t£ 
F. K^] BJ P. apo] sapra F. HP] PiV p. iarir 

[aav p. Tfaanpo] om p. to] om. p, td B. 26. JK] 

rx Pp. rj] in ra», Vi JCd Pp. imiv] lati Bpj «fa» V. 



s 

L 



140 ETOESEiaN p'. 

eaQU a^a T£t^aaliioia iaxi tov FK. atdktv iitsl Hev} 
iuzlv ij FB r;} Bz/, aXXa ij ftJv B^ r^ BK, nout- 
^ffri T^ rf/ Tffi;, ij iJ^ rS r^ HK, tovziett r^ if/7, 
^ffTM' fff?j, xc! ij Ti/ «pa t;; i/i7 iffTj fffitV. 3£«i iitel 
6 fffij ^eriv 17 ftiv ril rj; i/i7, ij Ss UP r^ PO, fffow 
^ffri xal TO fiij' .^ff za MII, ro d^ i7,i rp PZ. 
«AAa TO JWii Tfi i7.^ ^Crtv iffov ' naffaalr]Qdn«ta yufi 
zov MA nttQaXXtjXoyQajifLov xal to AH UQa za PZ 
iffov ^ffTtV- za ziesago: aga za AH, MU, TIA, PZ 

10 i'aa «A/ijAotff iStiV la tdaaasftt S^a zov AH iszi 
zftpanXaeia. ideix&t] di xul za zieaaQtt za VK, KJ, 
HP, PN tov VK tETQaziaeia " ta «pa oxtto, u xsql- 
BXit zov JlTT yvmfiopa, tetQttnldaid iazi zov AK. 
xaX iiiil zo AK z6 vnh tcov AB, B/J iettv i'ei] ya^ 

16 ij BK TjJ BA ' t6 apa zBTQdxig vxo tmv AB, BjJ 
TSTQajtldeiov iezi loiJ AK. iSsCx^ Sh zov AK ze- 
rgaxXdaiog xal 6 STT yvtoiiaV tb asfa tEt^axi^ 
vno tmv AB, BA taav ietl T50 ETT yviofiovi. xoi- 
vov XQoexeie&at To S&, iOTni fffov Tp airo t^? Af 

ao zsTQaymva' z6 a^a ifipKxts vjro zav AB, BA ntQi- 
Bxofuvov 6Q9oyc6viov ficza loij dzo AF TtzQaymvov 
teov iatl ta £TT yvwiiOvi xui iro S'®. dkXd £TT 
yvdiiav xal to S& oAoi' ^ffrl lo AEZA ttt^dyoivov, 
o ^flTtii dito r^s AA' ro aga rtzQaxis wiro rmv AB, 

26 BA jietd TOTJ ditb AP leov ierl ta awo Aj^ tstga- 
ydva' terj Si 7) BA trj BF. to «p« tEXQtixig vnh zav 
AB, Br niQifx6(itvov og&oyaviov piiza rou «»o AF 
Ttzpaymvov teov iatl za anb t^g AA, tovzieTi ra 
dnb t^gA B xal Srag dno fudg dvayQaipivzi TtzQayavei. 

1, iml laxiv PV; ilm p. 2. TB] BT F. alV F. 
BK] supra acr. iJ m. 2 V; mg. ^ BT oftt tj PH lativ foijV. 



i 



EELEMENTORirM LIBER H. 141 I 

r se aequalia aunt. ergo fl 

^K + FK-i- HP+ PN==4: rK. I 

us quoniain FB = BJ et BJ = BK = FH et ^ 

rB = HK=Hn, erit etiam rH^HTl. efquoniam 
FH = HO" et nP = PO, erit etiam AH = MH [l, 
36] et HA = PZ [id.]. uerum MH = HA- nam sup- 
plementa sunt parallelogrammi iWy/[I,43]. quare etiam 
AH = PZ. itaque quattuor AH, MFI, H A, PZ inter 
seaequaliaanut quare^H+JVr77 + ;7^+PZ=4^i/. 
scd demonatratum eat etiaoi 

rJC" + ffz/ + i/P + P^ = 4 rff. 
e^o octo apatia gnomonem ZTT efficientia = 4 AK. 
et quoniam AK = AB X BJ (nam BK = BA), erit 
A AB-XBA = 4: AK. aed demonstratum est etiam 
£TT = 4 ^K quare ^ ABxBA = STT. com- 
mune adiiciatur ^®, quod aequale eat AF^. itaque 
AAB-XBJ + Ar- = ZTT ■{■ S&. sed 

jjrr+s^e^v^EZ^^-^-^*. 

sque 4^Sx BJ + Ar* = AJ^. sed BJ = Br. 
ique 4 ^SxBr+^P' = ^^* = (^S + J5r)*. 




foi]] PP, iVij ^flitV B, Iflrii' iVk p et in 
n HiT fflii ^oci" mE. m. 2 V. TOt.i((r«» 
V Cet} Vp. ^di^f] (alt) laxl B, fi. ^or/j- PV. 
9. ^oni- iW Vp, iai^»-] F; ioU PB. 
«] (alt,) id P. 10, iffri"*] (fflt V: Ux! B. ■tEieanlBOH« 

lori zov AH p; loS ^H Ttx^anXaem ietir P, 12, « «£(?!- 

I^^ot p; KXE^ #2» F. 13. yvaiiiova xa F V, ion] ^otip 

Ti om, V. AK iauv V. 14. «Tto] «ird F. BJ] BK P. 
r yae] yue XKiV. 16. Bffj .KB P, 16. ^ortf PV; om, B. 
BfA iniiv B, mpaftliiafiDv p, 18, foitV V. tiu] corr. ex 
^ m. 2 B. 21. ^ r] PB, F m. 1 ; xfif A T Vp. m. 2 F. 

'88. iaxit FV. rro) (a!t,) corr. e» rd F. alk' F. 23. 

I«fr PPV. 25. >r] rn; ^r p. JorC* V. A^^ xijs 

Ad Vp. 37. Brj B.d B, corr. m, 2. AT^ i^s -«r Vp. 
•9S V' S8. Iffifv FV. looWociv V. S9. im^] om. p. 



142 ETOIXEIiiN |5'. 

'Ettv «pa EV&€ta yguftfii] Tfirj&^, mg iTv%EV, 

TSt^dxLs vao t^s oAtjs xctl ivog tmv r^ijftaTiav attQi- 

tXOfiBvov OQ&oymviov fiszcc tov aao zov Aomoi' Tfiij- 

[lUTog TtTpaytjovov iOov iazl tm ano ts rijg SAtjs xal 

^ rov BiQTjfiivov tff^fiatog ats anit (iias avayifagidvri I 



'Eav Ev&sitt y^afifi^ T^ijd'^ slg taa xaX 
aviea, Ttt ttjio Ttav avieav t^g o^tjs Tfiijfiatiov 

10 tizpayoivtt Sialdaia iaxt toiI xb kjto t^s ijf^i- 
aeiag xal rot) a%o t^g (lettt^ii Tmv rofttdv tetffa- 
ymvov. 

Ev&ita yaQ Tig ^ j^B ritiiiija&ta sig {i}v iaa xatk 
To r, tig dl aviaa xata to ^d' Xdym, oti ta aico rmy, 

16 A^, dB TBT^ttyava SiTrXaaia ictt tmv ano tiov AV^ 
Fz/ tBTQaymvmv. 

"Hx&a y«p «reo Tow r t^ AB n^bg OQ&ag i] FE, 
xal xtis&m foij fxtttaga tmv AV, FB, xal int^Evj^- 
&maav ai EA, EB, xal Sia filv toi d t^ EF na^- 

20 dkXriXog ^x&ta t} jJZ, Sia Si rov Z t^ AB ij ZH, 
xal inB^Bvx^a ^ AZ. xal inel lati iatlv ij AF rp 
VE, fffi; iatl xal {j vno EAF ymvia rfi vno AEF. 
Mttl inBl OQd^ iativ ij ngog t^ F, koLnal ap« «f vxh 
EAF, AEF fiia opdjj i'aat eiaiv xai siaiv [aaf rifii- 

36 9Eia aga opS^j iCTiv ixazBfftt twv v%o FEA, FAE, 



1 

1- 

I 



1. iav aga — 6. tciparcufnjj om. p. 1. itvje V. 2. »• 
ifttxie] mg. m. 2 V. i. iexCv P. aito te] « aiio PBV; 
ano F. 6. TtfOiiiiiiuivov P. 9, tlt KViatt p. 10. ioTiv 

FV. tf] poBtea add. m. 9 P. fiiuatias} can: ex futa|v 

m. S F. II. «ffl «>« uxo i^e fttialiS] om. P; oorr. m, rec, 
sed eQan. 15. lariv T. a;tD itov] om. F. IS. iiov] ia 



1 



IELEMENTORUM LIBER H. 143 H 

Ergo si recta linea utcunque secatur, quadruplam V 

Tectasguluni tota et alterutra parte conipreliensuiii H 

Onm quadrato reliquae paitis aequale est quadrato in 1 



tota simul cum parte nominata deseripto; quod erat 
demoQstraB du m , 

]X. 
Si recta linea in partes aequales et inaequales se- 
catnT, quadrata in partibus inaequalibus totius descripta 
duplo maiora sunt quadrato dimidiae cum quadrato 

tpsctae inter sectionea positae.') 
£ nam recta aliqua^B in aequa- 

/Q\z ^^^ partes aecetur in F, in inae- 

/_^.----Fn\^ qualea uero in jJ. dico, esse 

ducatur enim a F ad rectam AB perpendicularia 
PE [I, 11], et ponatur aequalia utrique AF, FB, et 
ducantur Ej4, EB, et per ^ rectae EF parallela du- 
catur ^Z, per Z autem rectae AB parallela ZH, et 
ducatur AZ. et quoniam AF = FE, erit etiam 
LEAr=-=AEr [1,5]. et quoniam angulus ad T situs 
jctus est, reliqui EAF -\- AEF uni recto aequales 
int [I, 33]. et sunt aequalea. itaque uterque angulus 




n&. FV. rS] B e 

20. AE] PBF; AB aeijdlXyiXat ^i»ia Vp. 

(laonn. 4—5 litt). 29. iatiT] htiv PFV. ' EAr] B 

mvta, acr. m. 1 V. ■/a,v{a[ om. p. ^Er] TE,* p. 23. 

«ij ro F, corr. m. 2. 84. flvCv] (priuB) i^o^BVp. 26. fxB- 

■«^K (in ras. V) a|ia imv vffo ^ET, EAF fjiiCaiid iartv 6q- 



i 



144 ETOIXEIfiN p'. 



Ifia ta avTcc dt} xal ixarsQa rwv vxh FES, EBF 
^(liOEia i6ziv oQ^i ' oXt] KpK ij vito j4 EB opflTj 
ietiv. xal iTtsl 17 wto HEZ ^fiiaBid ieriv 6^9i}s, 
opS^ 6h f] VIC0 EHZ- fffjj }'ap iffTi ry ivrog xal 
G datvavziov tij wnro EFB' Xotni} aQa tj vjth EZH 
i}(iiaaid iativ opO^^g" fffij «pa [^ffili'] ^ waro HEZ 
^avCa rij vxo EZH- Sert xal nltvQK ij EH ry HZ 
iettv fffTj. ndXiv inel tj wpog t^ B yiovia ^(tietid 
ieviv dpS-^s, op#^ #i 17 iwo Z^B" ftfj/ yap w«A(j/ 

10 ^UtI TTJ ^viAs Kai dnevavtiov ttj vjto ETB' AotJriJ 
fipa ^ W3E0 BZ^ Tjfiiasid ietiv op^&^g* fflij apa 5) 
apos tai -B yeyvia r^ vxb /4ZB' Sste xal altvpa r) 
Z^ xIevqu r^ z/B ietiv fmj. xal iael foij ierlv ^ 
Ar tfi TE, teov iorX xal ro dah AF ta ano FE- 

15 rd «pa dao riav AF, VE rerpdyava 6ialdatd iert 
row dito AF. totq di dxo rmv AF, VE taov iorl 
ro ano tijg EA TitQayiovov op#]j yap ij vno AFE 
ytavia ' 10 apa am f^g £-<4 dm^affiov ieri tov uno 
T^ff .^r". %dXiv, iatl Caii ierlv ij EH tfj HZ, ftfov 

20 xal ro aiEO r^s £i/ tp awo T^g /fZ ■ ra apa aKO 
nav £H, HZ rsrQdyfava dLTtXdeia ieti rou ano f^s 
HZ Tttifaytovov. rotg Si dao rmv EH, HZ tttffa- 
ytavoig teov ietl ro ajto tijs EZ tEr^dymvov ro apa 
awo r^s EZ rstQdywvov SinXdeiov iert roiJ aao tijg 

25 HZ. fffjj Si 17 HZ tfj r^ ■ ro apa ano r^s £Z *(- 
jt^afftov ieri toii ano r^g Fi^. leti d'i xal to dito 
T^S EA SiTtXdeiov row dno r^g AF' td «pa a«6 
rpw v^£, £Z Terpcyoj/a dLnXdeid iari rmv dxo riiov 

1. Slb Ttt — 2, oeff^e] mg. in ras. V. 1. ujtti] aupra iu.2 
F. EBr, FEB p, i. ^oti* P; comp, aupra V. 5. a5if»'a»i- 
Ti^ns p. 6. iaT^v} om. P. 7. EH] HE p. 1^] nXcuQa 1^ 
Vp: nlcSpf: add, m^. m. 1 F. 9. ncfliv idr/] itm aaXiv P; In/ 



1 



< 



^V^ ELEMENTOBUH LIBER U. 145 ^| 

^^frSAf FAE dimidius recti eat. eadem de causa etiam H 

^B uterque angulua FEB, ££FdimidiuB est recti. quare I 

^B L AEB rectus est. et quuutam L HEZ dimidius est I 

recti, rectus autem eet EUZ (uam aequalis eat 
angulo interiori et oppoaito EFB [1,29]), reliqaus 
i EZH dimidius eat recti. ergo L HEZ = EZH. 
quare etiam EH = HZ [I, 6]. rursus quoniam an- 
guIuB ad B aitua dimidius est recti, angutua autem 
Zj}B rectua (nam rursus augulo iuteriori et opposito 
MVB aequalis eat [1,29]), erit reliquus augulusBZ^ 
dimidius recti. itaque auguius od B situs aequalia 
est angulo ^ZB. quare etiam Zz/ = ^B [I, 6]. et 
quoniam AF = FE, erit etiam AF* = FE^. itaque 
AI^-^-rE^^i jr^. sed ^^^^^r^ + rE^Cnam 
t-^-TErectus est) [1,47]. itaque£-^* = 2^r'. ruraua 
quoniam EH = HZ, erit etiam EH^ = HZ^. quare 
EH' + HZ' = 2 HZ'. uerum EZ^ = EH^ + HZ* 
[1, 47]. itaque EZ- = 2 HZ\ sed HZ = FA [I, 34]. 
it»que£Z'=2rz/*. uerum etiamfi^^^^^f. itaque 
^£' + EZ* = 2(.4r*+r^').3ed^Z' = ^E= + £Z' 



aupra F. 11. BZJ] ^ZB P. 12. JZS\ BZJ p. 13. 

Z^] PF; JE BVp. U. ievC] om, B, aupra P. AT] 

PB, F m.Ii i^s^r Vp, F m.B (r-4, Bod corr,). TE] x^t TE 

I Vp, F ta.2. 15. Ta «(« ajta rav AF] TiTgajavov seq. lao. 

rS litt. ip. rmv] i^s comp, p. Iotiv V. 16. AP] rijs 

WAr Vp. F m. a. ioTir FV. 17. irii om. F. E^J JE 

FPp. 18. aao] v«6 qj (non F). EA} JE P etV m. 1. 

fjfln» PV. 19. r^s] om. P. EH] m rae. V. fffo»] 

PBF; hov karf Vp. 30. EH] HE P et F, sed corr. 81. 

iativ V. 33. iaxi] eupra V. itie^r^^ojj P^l o™' BVp. 

24. m9ciy(a*>ov] pnnotis del. P. leiiy V. 25. HZl Z 

in rae. m. 2 V. tar, Si — 26. TJ] mg. m, 2 V. feri Si ij 

HZ T^ r.<| aXia to ano x^s HZ roof iatl -cm a«D irg FJ P. 

16. fow* V. B7. E.J] in ras, V; AE p. 'rou] ^a« (comp.) 

xoi <p. 88. ,*£] inter A et E raa. 1 litt. F. ^trrit' V. 

edd, HniboTE sl filcage. 10 



I 



r 



146 



STQJXFJflS p'. 



j4r, r^ z£t^ay<6v<ov. rofg Sl «iro rtov AE, EX Heov 
itfTi. ro ano r^s j4Z TET^ayiDvov ' op&ij ydp iOTiv 17 
vxo AEZ yavUf ro apa «jto t^g ^Z TfEpaytovoi' 
Si-xkaetov iezi rav aao xmv AF, F^. ra di «wo 

I T^S -^Z Tffft ra «wo rmv A^, AX' opft^ ^'ap 17 wpos 
rt5 ^ yavia' ta &Qa ano tmv AJ, JX Si^Xdeitt 
iuti tmv dito tmv AF, Fid nTQaydviav. i'Gi} Hl 15 
zJZ t^ ^B' Ta «Qa dm tmv A^, ^B tst^dyiavK 
Si%kd6id ieti zSv dno riav AF, F^ Tittfaydvav. 

) 'Eav apa tvQ-tta yQaii[ijj Tfi^Q-fj sis ^f^ ^"■^ avica, 
Ta unb ttav dviecov r^g oAjjg T(i7)(idt€3v tatQayava 
Si7ti.d6id iatt Tov ts dno T^g tifiiaeiag xal roii dxo 
r^S fLiTaiv Tav toiiwv TctQaydvov " oH£p iSti dfC^ai. 



} 'Eav evQ^eia ytfufifir] rftijfrjj Si^a, stffoOTE&^ 
8i ttg avtij av&Bta in' tvf^tiag, t6 dxb T^g 
oAijff avv tij xffoexstfiivrj xal tb dxb rijg xpoff- 
XBifiivTjs za <}vva(iip6t£pa tttQayava SiaXdeid 
iati zov TB dnb z^g iifiietiag xat zov dno 

3 r^S evyxttfidv^S ix ts r^g •^ln.eEiag xal z^g 
nQoexEifiivtjg a>s rko tiidg avayQayivzos ts- 
ZQaydvov. 

Ev&tta yap rtg rj AB tsrfHjff^fi) Si%a xata ro T, 
nQoexEie&a Ss ttg avzy sv&ila ia EV&Eiag {] B/1 " 

& kiya, oxt ra dnb Ttov AA, z/B tEZQdyava Stnkdeid 
iert Tmv d«b Tmv AF, Fj^ TSTQaydvav. 

"Hx&a yag &«b zav T e^fiEiov tjj AB jiQog oq^&s 



i 



2. Inlv V, ztxt^yavov] om. p. 
m, 1 r. 4. iitiv V. tMf] (alt.) trle BP 
AZ] corr. ei AZ F. 7. hziv FV. 




ELEMENTORUM LIBER U. 147 

(nam AEZ rectus est) [I, 47]. ergo 

Az^ = 2 (An + rj^. 

uerum Ajd^ 4- ^Z^ «= AZ^ (nam angulus ad ^ situs 
rectus est). itaque AJ^ + JZ^ = 2 (^F* + F-J^). 
uerum ^Z = ^B. itaque 

A^ + JB^ = 2 (^r» + r^«). 

Ergo si recta linea in partes aequales et inaequa- 
les secatur^ quadrata in partibus inaequalibus totius 
descripta duplo maiora sunt quadrato dimidiae cum 
quadrato rectae inter sectiones positae; quod erat de- 
monstrandum. 

X. 

Si recta linea in duas partes aequales secatur, et 
alia recta ei in directum adiicitur^ quadratum totius 
simul cum adiecta et quadratum adiectae simul sumpta 
duplo maiora simt quadrato dimidiae et quadrato rec- 
tae ex dimidia et adiecta compositae.^) 

■p 2 ^^^ recta aliqua AB in. duas 

partes aequales secetur in Fy et alia 
^ recta iS^ ei in directum adiicia- 
tur. dico, esse 
^ AJ^ + JB^==2{An + r^^). 
ducatur enim a puncto F ad rectam AB perpon- 

X. Boetins p. 386, 7. 




1) (2a + 6)» + 6« = 2[a«-f (a + 2^)«J. 



S. JZ] Z m ras. V. 9. icuv V. 12. iaziv V. row] (alt.) 

add. m. 2 V. 18. ta] om. F. 19. iaziv P V. 20. ts} 

insert. m. 2 F. 21. avctyqatpivxi, tSTQaytova) P. 26. 
icxtv V. 



10» 



148 ETOIXEiaK p'. 

7) FE, xkI xtitf&a i-aij ixKTspa tiav AF, fj 
i'jiEi,Bvx&a<fav al EA, EB' xal diic ^iv toi E Tfj 
AA TtapaXl^^Xos iJzO-o rj EZ, dia Si toi A rfi PE 
TtaQK^Xjjkos ijx^io ^ Z^. xal imsl eig aapaXl-^lovg 
5 iv9s(as r«s EF, Z^ E^ffara Ttg ^viitsrftv ij EZ, al 
uiro FEZ, EZA apa Svslv oQ^atg tOai. EleiV aC 
ttQa vTtoZEB, EZjd Svo OQ&mv si.dasovi? sltfiv at\ 
d^k U7t ilaecovmv fi Svo OQ&mv ix^aXXoftcvai ffvfiaC— 
nzovatv ttt oLQa EB, ZA ix^aXXofisvai ijtl za B, A 

10 y-sQyi 0v(in£aovvrai.. ix^s^X^^a&aaav xal avfixixtdTto- 
aav xata ro H, xal ijts^evx^o) ij AH. xal insl fff»j 
iatlv 71 AFt^ FE, taij iatl xal yavia ij vxo EAF 
tfj vnb AEF' xal OQ&ij ri nrpoj ra F' Tjjiiasitt «pa 
OQ&ijs \iativ] ixatepa t^v vito EAV, AET. Sia ra 

16 a^ra iJ^ xal sxareQa zinv vno FEB, EBF iiyiCasitt 
iariv opff^s" opS^ apa iatXv tJ vno AEB. xal inel 
fiftiasia oQd-rjs iartv tj vnb EBT, rifiiasta «pa OQ&ijs 
xal ij vnh ABH. iatt Ss xaX r\ vnb BAH opd'^* 
fff») yaQ iati T^ vnb AFE' ivalXa^ V^Q' ^otrcij apa 

20 7] vnb AHB rtiCasia iariv opfl-^j' ij «pa vnb ^JHB 
T^ vao A BH iettv fijij ' 5axE xal nXsvQa r) B^ 
nXsvQtt t^ HA iativ farj. naliv, insl rj vnb EHZ 
rjfiiasid iariv OQ&ijg, OQ^ri S} ij jrpog rra Z ' /Oij yuQ 
iatt t^ aitsvavtiav tfi XQog rp V' Xotnri aQa t] vno 

86 ZEH rfiiacia iativ dpft^s' tarj apa fj vnb EHZ 
yavia tj] vnb ZEH' rotfze xal nXsvQa f HZ nXsvQ^ 

3. tav J in r£] tov ^ FE gi. TE] PE ndliv P. 

4. Z.J] PF; JZ BVp, B. EF, ZJ] in ras. V, rS, dZ p. 
7. ZEBl in raa. m. a F. EZifl J in raa. V. ll.aTrmsB 
p. 8. t«7t'] PVi a^o BFp. 13. iat{v PV. EAT] PB, 
in ma. V; ^ET p, ia raa. F, 13. AET} PB, in ras, V; 

EAT Fp. 11. loTiy] om. P, aupra F. IS. AES] EB et 



I 



\ 



I 
I 



ELEMENTORUM LIBER II. 149 

dicularia FE, et ponatur utrique AF, FB aequalia, 
et dueantut E^, EB. et per E rectae J^ parallela 
ducatur EZ, per ^ autem rectae FE parallela duca- 
tur Z^. et quoniam in rectas parallelas EF, Z/i 
recta aliqua iucidit EZ, anguli TEZ ~f- EZ^ daobus 
rectis aequates auut [I, 29]. itaque ZEB-\-EZ^ 
duobus rectis minores sunt. quae autem es augulis 
minoribus, quam suut duo recti, educuutur rectae, con- 
curruQt [«fr. 5]. itaque EB, Z-J ad partes B. J edue- 
tae concurreEt. educantur et concurrant in //, et du- 
catur.rfif. et quouiam AT = rE,eiitL EAr=AEr 
[I, 5]. <^t augulus ad F positus rectus est. itaque 
oterque angulus EAF, AEF dimidius eat recti [I, 32]. 
eadem de cansa etiam uterque augulus FEB, EBF 
dimidius est rectL ergo L AEB rectua est. et quoniam 
L EBF dimidius recti est, etiam L •JBH dimidius est 
recti [I, 15]. sed LB^H rectus e»t; oam aequalis 
est angulo ^FE (aJternus enim est) [I, 29]. itaque 
qui relinquitur angulua ^HB dimidiua est recti. erit 
igitur L ^HB = JBH; quare etiam BJ = H^ 
[T, 6], rursus quonism L EHZ dimidius recti est et 
angulua ad Z positus rectua (nam aequalis est oppo- 
aito anguto ad F[1, 34]}, erit, qui relinquitur, angulua 
ZEH dimidius recti [I, 32]. itaque LEHZ = ZEH. 
quare etiam HZ = EZ [I, 6]. et quoniam 



inter haa Utt. I litt. eras. F. 17. 090] afa laiiv p et eupra 
P. 18- Miv V. *ai] om. p- 19, ioriv V. ynp] 

supra m. 2 F. 20. zfHB] iJBH V, eorr, m. 2. iiiiietiK 

— JHS] ora- P. JHB] litt- HB e corr. V- 21. JBH] 

H e corr. V. (■trij itxtv p. Bd] JB p. 32. H-Jl JH 

Pp. 24. huf PFV. 25. EHZ] ZEH p. 26- ZEH] 
EHZ p. HZ] in ras. 111.2 V; ZE p et F in. 2. 



1 



160 STOIXEKN f. 

zfi EZ dariv lerj. xal ixBi ^leti ietXv ij EF tjj r-rfj 
(fSov iexl [x«!] th ««6 r^s EF Tetpaycjvov rra «wo 
rrjs rV/ tftgaycova ' ra «pa as^o Tioi/ EF, F^ wrpa- 
yova diitldetd iffri tou ttjro r^s FW TfrptEyrovow, 
5 Tolg de uno tmv EV, VA teov ietl t6 axo r^s EA- 
t6 aQa KWo r^s EA TtTQayavov SiitlaSiov iazi tou 
Kjro rfg AF Ter^aysavov. noif.iv, inil fcij ierlv rj 
ZH rfi EZ, teov ierl »al rh d.nh i^g ZH 
T^S ZE- ra UQa ano rmv HZ, ZE dialdaia iexi 

10 Tou aTCo rfjg EZ. Tofg di dnh tmv HZ. ZE iGav 
iaxX ro dsth rtii EH- rh aga «reo x^g Eti SntkdeioVQ 
iexi roii KJio T^s EZ. ftfi; Se ^ EZ t^ F^' xo apici 
«jro r^; EH nTgdytovov SiTtkdatov iaxi xov airo r^gi 
r^. iSe^x^V ^^ '^^^ ^^ ^^^ ^VS ^^ Stxkdetov xov> 

15 «OTO T^s ^F' T« apa <wro riav AE, EH xtxgdyfava 
dtnkdeid iaxi rmv «ffo rtoi' AF. FjJ rBTpaytovav. 
TOig Si dxo Tiov AE, EH TergaytovotB i^aov iexl ro 
«Ko rijg AH xixpdyavov to aga dxh r% AH di- 
xXdetov iexi xav djtb xtav AF, FA. xa de dxh r^g 

20 AH lea ierX ra dnh rtov AJ, ^H' ra aQa awo 
rmv A^, iJH [TfTpwyova] Smldatd iart xmv dxo 
xmv AF, r^A [rerQayavmv]. tai] Ss xj ^H r^ idB' 
rd dga dnh rav A^,^B [Tfipaj-rora] StxXdaid itfri 
rmv dnh rmv AF, FA rfrgaymvmv. 

35 'Eav UQa ti&tta yQUfifiii rfirjQ-ji Slxa, XQoere&^ Si 
Ti^ awTjj {vQtia ix £v9e{as, x6 dno rf^s oAijg ffw 
xfi «QoexetiiivTj xaX ro ano r^g aQoexEifiivr]g rd ffw«.. 
aiuporeQa TerQaymva SixXdeid ieu rov xe ano xijf^ 



I 



1. EZ] ZE P; ZH p et P m.2. 
rA] om. P. EF] AFv. FA] in 

2. iaiiv V. xtti] om, P. i^j] om. 




ELEMENTORUM LIBER H. 151 

En = r^, erunt En + F^* — 2 FAK sed 

EA^ — £;r« + FA^ [I, 47]. 
itaque EA^ = 2 AF^. rursus quoniam ZH = EZ, 
erit Zfl* = ZE*. itaque HZ* + ZE^ = 2 EZ«. sed 
EH^ = HZ« + ZE^ [I, 47J. itaque £H* = 2 £;Z*. 
uerum EZ = FJ [I, 34]. ergo EH^ = 2 r^». et 
demonstratum est etiam EA^ = 2 ^r^. itaque 

^£;« + EH^ = 2 (^r* + r^*). 

sed ^H* = ^£:« + EH^ [I, 47]. itaque 

AH^ = 2 (^r* + rj^), 

sed ^H^ = ^^« + JW [id.]. ergo 

^^« + ^H^ = 2 (^r^ + rj% 

uerum ^if = ^B. itaque 

A^^ + ^B^ = 2 (^r^ + rz/«). 
Ergo si recta linea in duas partes aequales seca- 
tur, et alia recta ei in directum adiicitur, quadratum 
totius simul cum adiecta et quadratum adiectae simul 

V; AF y. TSTQixyaivov] om. p. 3. FA] FE p. tstqu- 
ymvtp] om. p. AF, FE p. TSTQciycDva] om. p. 4. FA] 
corr! ex AT Y; AV p. 6. ET, FA] AT, FE p. EA] AE 
P; AE TSTQaymvov p. 6. Tijg] tAv F. EA TSTQocymvov] 

AE p. SGTLV V. 8. ZH] PF, V m.2; HZ B, V m. 1 ; 

EZ p. EZ] ZE P; ZH p. ZH] HZ P, EZ p; ZH 

TSTQdymvov V et m. 2 F (comp.). 9. ZEl ZHp, ZE 

TSTQaymvtp V et F m. 2 (comp.). HZ] PF, V m. 1; ZJf B, 
V m. 2; EZ p. ZE] ZH TSTQoiymvoc p. ioTiv V. 10. 

EZ, ZH p. 11. EH TSTQCcymvov Vp, comp. supra F. 12. 
iaTiv V. 13. TSTQciymvov] om. p. sotiv V. 14. EA] 

corr. ex E-:^ m. 1 P; ^E p. 15. ocqoc ano] qp, eeq. -no m. 1 
(del. q>). EH] HE F. TSTQOcymva] om. p. 16. sOTtv V. 

TfT^ayoovoov] om. p. 17. TSTQaymvoig] om. p. sct^v V. 

18. T«T9ayQ)»ov] om. p. 19. sctiv V. 20. sotiv V. 

21. TeTpcfyoova] om. P. SinXdaiov qt (non F). sotiv V. 

22. Fz/J in ras. V. TSTQaymvmv] om. P. 23. xsTQdymva] 
P; om. BFVp. saTiv V. 26. oUiys 9. 27. t6 a«dj 
om. PB; m. 2 insert. F. 28. sctiv V. 



I 

I 

k 



152 ETOIXEBiN p\ 

'^fiieetag xal tov aith r^g evyxdfitviig ix « tjjg ^ 
atiag xal rije nQoexitfiivrig wg axo ftt«s «vaypKyEitoj 
rsTpaj-iBvov ojtfp Hbi Set^ai. 



I r^i/ 8o&iiaav ev^^etav rtfietv meti to vx^M 
r^S oXijs xfti Tov {tsqov trav t[tti(iatmv xe^iM^ 
tx6[ievov op'9oj'ro'vtoi' fffov frvat rp k«o to« 
Xoixov rftTJfiaroff rfrpayrajJCJ. 

"Etfrra i) So^etaa sv&eta ri JB' Set dij r^v AB ' 

10 xsfietv iSffTE ro t>3ro zijs oXrjs ««i ^ow ertpou raiv rfHj- 
/iKrrar W£p(f]:o'ft£voi' dpftoyraftoi/ tsov itvai rp aai J 
roti ^otHou ifti^ftaros rstpaymi'». 

'^uayEypaqcitfo j-ap awo r^g ^B tjrpaj'roi'oi' tAI 
AB^r, nal rfrft^Odro ^ -^F #^j;a xar« ro £ ( 

15 (ietov, xal ine^evx&a 7) BE, xal dtjjj;*to rj FA ixl 
ro Z, xol XE^e#io rij BEterj {} EZ, xal oKaj-fypoyd» 
aJto T^s -4Z r£Tpaj'cavoj' ro Z®, xai (J(ij'x*ra ^ H& 
ial To ff ■ i^j-to, ort ^ ^B r^rftijrai xoTa ro ©, tSffrc 
TO lOTo Ttoi' AB, B& jie^iexofiEvov dp^oj^oJiftov fffov 

20 jtoielv Tto njto r^g -40 rtrftcej^tiji/fo, 

'fiifl j/ap ev&eia ij AF rerfitjrai, Si%a xaxa ro E^ j 
^rpdffxftrai Se amfi tj ZA, ro apa vno rav VZ, ZA I 
aeQiex6(ievov 6p9oyiaviov (tfra roi) ajro r^g AE ts- 
t^aymvov fUov ierl tm n;ro r^g EZ tetffayiDva. fffij 

S6 tfi 7) EZ rfj EB' ro «pa vxo tmv fZ, Zj</ ^eta \ 
xov «JEO T^g AE rffov j^Orf rto an^o EB. alKa rto (i«6 ' 



2. («Taypaqi^fTOS TtrpayaJroi)] cocr. ei dvay^aifiifti Hip»- 
yaSvgj m. l P. Prop. XI cnm praeeedenti coniuniit V; corr, 
"' "" " ■", ni, 3, 6. •00»' fv&ii- in raa. p. 6. Tjtij- 

3 litt. V. 8. TrTpayoSfoo F. 14. A&^r'^ 



I 



■ ELEMENTORUM LIBER n. 153 

Bimpta duplo maiora mnl quadrato dimidiae et qua- 
drato rectae ex dinitdia et adiecta conipositae; quod 
erat demonstrandum. 

XI. 
Datam rectam ita secare, ut rectangulum tota et 
Eklterutra parte comprehensum quadrato reliquae partis 
aequale eit. 

Sit data recta JB. oportet igitur rectam JB ita 
secare, ut rectangulum tota et alterutra parte com- 
pFehenBum quadrato reliquae partia aequale sit. 

coiiatruatur euim in .•IB quadratum JBJF [I, 46], 
et AP in duas partea aequales secetur in puncto E, 
et dncatur BE, et FA ad Z educatur, 
et ponatur EZ = BE, et construatur in 
AZ quadratum Z8 [id.J, et educatur H& 
ad K. dico, rectam AB ita sectam esae 

in ©, ut faciat ABxB&= A®\ 

nam quoniam recta AT in duas partes aequales 
secta est in E, et ei adiecta eat ZA, erit 
L rZxZA-\- AE^ = £2* [prop. VI]. 

pied EZ = EB. itaque FZxZA ■{■ AE^ = EB\ 

_ XL Boetins p. 386, 16. 

ABVJ B, AB, inBerfdB TJ m. 8 P, AFdS p. 17. ZS] 

ZHeA p; in FV poet Z et pOBt B 1 litt erae. di^x«in] 

*i- sopra m. 3 F, 20. jroiEi>] PF; thai Bp et pofit ras. 3 

litt. V, xai\ mg. m. 2 p. 24. /oii'] comp. Bnpra m. 1 V. 
ojid] tp, aeq. iio m. 1. EZ] in rHE. F. 26. FZ, Z^] 

in ras. F. seq. dfid^ayaiviav qj, quod cum eeq. ^citi in mg. 
tranEit. firta] PB et BiDe dnbio F m. 1; nt^itxafi.evov op- 

9oyaviov ficra Vp, et P^. 2. 2«, «ko r^s] om. P. AE 
ttxeayavov Vp, F m. 2. ietiv V. _ EB) PB, r^s EB P, 

itTeBytovai add, m. 8; tijs EB xtteayiovm Vp. 



1 



154 

EB ra« £0x1 ra , 
rp j1 yavi 



ETOIXEUJN p'. 



'«liM 

KQt- ^ 



co tav BA, AE' op&^ jwp ij 
«ea lOTO Tiav /"Z, Z/4 f^Era 
«Jto T^s ^iT /■tfoi' ftfrl rots ano rmv B^, j^£. 
vov aqwjpjjffffo ro a«6 r^j ^E' i.oixov apa r 

6 raiv rZ, ZA nsQisxofiBvov 6ff9oyc6viov teov iotl rto 
Kjto T^s ^B T£Tpay(6va3. xa{ iert zb fih' vao Tmw 
rZ, ZA ro ZK- tati ya^ ^ ^Z x^ ZH' xo ds aito 
Tijs ^B t6 A^- ro npa Zif fffoi' ^ffri rta A^. Jeo(- 
voi' K^wjp^eS'» r6 AK' i.oi7tbv apa ro Z® rra ©^ rffov 

D seriV. xai ieri ro (i.%v&jJ tb imo tcovAB, B&tatiyag^ 
AB zfi B^' To 6i Z& t6 awo Zfjg A®' to apa uKo 
zmv AB, B& TtfQtsxoftsiiov oQ^oyaviov toov iiftl r^ 
ano &A Ttvgaymva. 

'H opo do9-£t0a cC&sI^a ^ AB Tttfitjzai 

3 @ mtJTs To rffo Tcow ^B, B& xcQiixofisvov dpOoyfl»- 
vtoi/ fffov xoiBlv ta a«b Tijg &A tSTQtcysava' om 
idst aoiTJeat. 

'Ev Totg ttfi^Xvytovioig rQiyeovoig zb a«d T^jS 
T^v dfi^Xstav yaviav vaoTBivovaiiq nlfvpajji 
BO tsTQayovov (ist^ov iati zmv djib tav rijv ufi- 
fiXstav yaviav aigisxovaav nXtvQav TBVQa- 
yavtav rra wep[£;i;oftt'vro dig vno re ynag 
TtsQl r^v dfi^lsiav yaviav, i^' ^v ^ xd^i 
aijtTBi, xttl Jrijs dTtoXafi^avoiiiviiis] ixzbs vs^ 
5 i^ff xa^^fTOV ffpog ty dyL^ksia yavia. 

"Eata d^^XvytavLov zpiyavov ro ABF dfi^Xstca 



^ 1. r^e EB Vp, F m. 2 (KB corr. ei E J). My Y. 

S. lariv V, comp. enpra F. 4. i^^.<£ «ipuymvov p. 6. 
riea'oj'ffli'iiw] om. P. ^or^i. V. 6. ^oti*- V. 7. AZ] ZA 
p. et V fled corr. m. 2. 8. ^ati'»' V. 9. SJ] J» B et V 



d^ 



ELEMENTORUM LIBER n. 155 

sed Bjfi -{- AEl^ = EB^] nam angulus ad J positus 
rectus est [I, 47]. itaque 

rZxZA+ AE^ — BA^ + AE\ 
subtrahatur, quod commune est; AE^. itaque 

rZxZA = AB\ 
et rZxZA^ ZK\ nam AZ — Zfl. et AB^ = AA. 
itaque ZK — A^. subtrahatur^ quod commune est^ 
AK. itaque Z© = ©^. et Q^ ^ ABX i5©; nam 
AB = B/1. et Z® =Ae^\ itaque ABxB» = SA\ 
Ergo data recta AB in 9 ita secta est, ut faciat 

ABXBS = ®A^. 
quod oportebat fieri. 

xn. 

In triangulis obtusiangulis quadratum lateris sub 
obtuso angulo subtendentis quadratis laterum obtusum 
angulum comprehendentium maius est duplo rectan- 
gulo comprehenso ab altero laterum obtusum angulum 
comprehendentium, eo scilicet^ in quod perpendicularis 
cadit^ et recta a perpendiculari ad angulum obtusum 
extrinsecus abscisa. 

Sit triangulus obtusiangulus ^iSFobtusum habens 



XII. Boetius p. 386, 18. 



e corr. m. 2. 10. iaziv] FV, icrr/ uulgo; iariv taov p. 

iczi] iativ V. Gd ro vno — 11. t^s A&\ ZQ xb dno rijg 
A& to d€ ©J ro vno AB^ BG P, Campanus; fort. recipien- 
dum. 11. AB] BA p. 12. iaxiv V. 13. GA] tijg 9A 

F, V {QA in ras.), trig -<4(9 p. 15. nBqiBx6ii,Bvov offQ-oyioviov] 
om. p. 16. noiBiv] PF; ilvai Bp et post ras. 3 litt. V. 

SA] va ras. m. 2 V; -<4© p. tBtqccyfovm] om. p. 17. noi- 
^ffat] dBi^ai p, corr. mg. m. 2. 20. iativ V. 22. xf] in- 

sert. m. 1 F. 23. riv] r^v iyL^Xrfistaav p, et B m. recenti. 



156 STOIXEIiiN p'. 



1 



£');oi' i^v v^b BAT, xal ijx&ci} ano rot' B aijfitiav 
inl Ti}v FA ^x^Aij&iraav XK9srog ij Bzl. isya, oti 
t6 «jto t^s BF tirgdyavov fift^ov ieri rmv ajio tmv 
BA, A r tBZtfayaviDv t^ Slg vm rav VA, A^ nep»- J 
5 fioy,ivai OQ^oy^viia, H 

^EnA yaff sv&sla f} Fd r^ftijTKt, (og hv%BV, narii' 
ro A <ti]nstov, To «pK «OTo i^g ^P teov ietX totg 
ano rav FA, Ad rtTQayiavois xal rra dlg vito rav 
VA, A^ Hfi^itfja^ivs» OQ^oyavla. xoivov n^oGxeie&ta 

10 t6 ««6 T^s ^S ■ TK Kpa «wo Ttov r^, ^B toa ietl 
tofg T£ areo Ttov f^, ^-4, AB titQuydvoig xal rip 
Slg vjto rmv VA, A^ [3r£pifj;0|i(s'i/p o^%oyxavia\ «AXa 
ToEff \]^v ano rwv F^J, ^B i'Gov iotl to an;6 r^g FB' 
op&^ yag ^ wpoff tw ^ ytovia- totg Ss ano tav A^, 

16 i^B teov tb ttjio t^g AB- to opa ax6 trjs FB tt- 
tpdymvov taov iiSrl torg ts anb rtav FA, AB tsrpa- 
ytovots xal rw dlg vno tav VA, A^ asQisxojiiv^ 
op&oytoi/t'^ ■ wtfre t6 axo r^g FB rsTQaycnvov tmv m 
dnb rav PA, AB tstpaytovav (itt^ov ieti ret dlg tix&ifl 

20 tav TA, AA xtfjisxo^ivia bff&oywvi^. T 

'Ev «p« rofg dfi^lvyaviois tptj/dvotg ro ««6 t^s t^v 

d^^Xttav ymviav vxotsivovaijg x^tvgdg rstpdymvov litt- 

£ov iuti. tav dnb tav t^v d(i^i.ttav yaviav xsquxov- 

amf nXsvQiov retgaymvav tp xtpiExo(iiva Slg vjcb 

■Ib rt fiidg rmv nt^l trjv dfi^lttav yaviav, iip' ijv t} 
xd&stog zintti, xal tije dnoXafifiavoiievrig ixtbg ujto 
r^g xafrd^cov npog rij d^^kBia ycavia' oxsff SSet Sst^ 



1. rii»] bia P. BAF yaviav V. 2. ^npiij&efto p. 

3, ^oii* V. 4. tffl*] om. B. 6, hvxf Vp. JF] T^ P 

et V m. 1. 8. im] rti»' V. S. ocfroyoinov V; corr. m. 2. 

10. JB] B^ F. ' (oiiv FV. 11. tftjaviD*ois] om. BF. 



JHk 



P 

2. 

P. 

i 




ELEMENTORUM LIBER H. 157 

angalum BAF, et ducatur a puncto B ad FA pro- 
ductam perpendicularis B^. dico, esse 

BT^ — BA^ + An + 2 r^ X A^. 
nam quoniam recta Fz/ utcunque secta est in 
puncto A, erit ^I^ = TA^ + AJ^ + 2 TAX AJ 
[prop. lY]. commune adiiciatur ^fB^. itaque 

FA^ + JB^ — r^ + AJ^ + AB^ + r^ X -^-^. 

j^ sed FB* = T-^* + -^B*; nam angulus 

ad A positus rectus est [I, 47]. et 

AB^ — AA^ + JB^ [id.]. 
itaque 
FB^ = r^ +^B« + 2ry^ X A^. 
quare quadratum rectae FJ3 quadratis rectarum FA, 
AB maius est duplo rectangulo rectis FA, AA com- 
prehenso. 

Ergo in triangulis obtusiangulis quadratum late- 
ris sub obtuso angulo subtendentis quadratis laterum 
obtusum angulum comprehendentium maius est duplo 
rectangulo comprehenso ab altero laterum obtusum 
angulum comprehendentium^ eo scilicet^ in quod per- 
pendicularis cadit, et recta a perpendiculari ad angu- 
lum obtusum extrinsecus abscisa; quod erat demon- 
strandum. 



12. «£^t820fi^v(» op^oycov^o)] om. P. 13. VA, A J q>. 

ietiv V. 14. AJ] rj V (non F). 16. tcov] PBF; tcov 

lctCv Vet p {htC). AB"] BA p. FB] BT p. 16. ictCv 
y. 18. tsTQdymvov (jLsiiov ictt p. 19. fisiiov icti] om. p. 
ictiv PV et B (y in ras.). 21. iv] idv qt. toiydvoig] 

om. P. 22. ymvCccv] om. P. 23. ictiv V. ano tmv] 

supra F. 25. ts] insert. F. ^y infiXrid-stcciv p. 26. 

ixtog] intos tijg tp. 




'Ev totg o^vytavioie tQtymvois to ajto Tij\ 
t^v o^etav yaviav vntorsivovtsrfs ^lBv^a 
TQttyavov £i.tttt6v iati Trciv dxo tav tiiv o| 
5 cZav ytavCav neQi.Bxovacov JiAivprov tEtgttyavojv 
T(5 nEQisxofiiva Slg imo ra iiiag tav «fpi r^v 
o^stav ymviav, itp' tjv ^ xa&sxog xiatst, xaia 
T^g aaola(i^Bvo(iivr)s ivtog vtlo tijg xa&ito^ 
«pos tfj o^Eia ymvi^. 

10 Eazcj o^vyavtov TQtymvov to ABV o^stav iiffiv 

rijv TCQOS ta B yaviuv, xal i^x&to anb toij A eijiteiov 
inl riiv BF xa&txoq t) AA' hiyw, Zxi tb anb x^g 
AT xtTQayiovov sXaxTov iuti rmv «ko tiov VB, BA 
tEt^aymvcav xa Slg vnb tav FB, B^ nsQttxo (livip 

16 oQ^^oyavi^. 

'Exsl yaQ sv&sCa rj VB tstfitjtai, lag ixvxsv, xati 
xb id , ta aga anb tav VB, Bzl XExQayava tOa iiSt\ 
rp xs S\g vnb tmv VB, BA jrjp(£j;ofi£Vra oQ&oyaivioi 
xal xa axb x^g AV xsxQayatvip. xoivbv nQoaxsiad-a 

20 tb anb r^s ^A xstQaycavov xa «Qa aitb rav VB, 
BiJ, .dA TstQuyava lUa iatl rro rs Sig vjib twv VB, 
BA asQtsxOfiiva oQ^oyiDviip xal ToFg dnb xmv A^, 
^V xstQaymvotg. «AA« roEg ^sv anb Troi/ B^, ^A 
taov rb aitb r^s AB' OQ^ri yaQ ij itQog ta z/ yoj-i 

26 vi^' totg Se aitb tmv AA, AV Haov xb anb r^g AV' 
xa KQtt dnb rwv VB, BA Haa ^Uri rra xs dnb xiji 
AV xal tm Slg vnb rtov VB, B^' aats ftovov 
dnb rijg AV IXarrov iaxt xmv anb rmv VB, BA re-' 
tQaytovtmv ta dlg vnb tav VB, BA asQtsxOfiiv^ 

30 &ayavCe>. 

4. li.ttaaoi' F. l«iiv V. 12. ET] B e corr. m. ■ 



m 



ELEMENTORUM UBER IL 159 

xin. 

In triaiigulis acutiangulis quadratum lateris sub 
acuto angulo subtendentis quadratis laterum acutum 
angulum comprehendentium minus est duplo rectan- 
gulo comprehenso ab altero laterum acutum angulum 
comprehendentium, eo scilicet^ in quod perpendicularis 
cadit^ et recta a perpendiculari ad angulum acutum 

intra abscisa. 

Sit triangulus acutiangulus jiBF acu- 

tum habens angulum ad B positum^ et 

ducatur ab A puncto ad JSP perpendicu- 

laris A^. dico, esse 

^ Ajy = rjj^ + BJ^ ~ 2 rjj X bj. 

nam quoniam recta rJS utcunque secta est in z/, 
erunt FB^ + JJz/» = 2 rJJ X JJz/ + z/I^ [prop. VH]. 
commune adiiciatur ^AK itaque 

rjj* + -Bz/« + JA^ = 2rBxBj + AJ^ + z/r^. 

sed AB^ = BJ^ -^ jdA^\ nam angulus ad z/ positus 
rectus est [I, 47]. et AI^ = AJ^ + JI^ [I, 47]. ita- 
que rJJ* + BA^ = AF^ + 2rBx BJ. quare 
AI^ = rjJ^ + BA^ -H 2r-B X BJ. 




XUI. Pappus V p. 376, 21. 



T^ff] om. P. 18. iXacaov F. ictiv V. x&v dno xmv'] 

zm vno F; corr. m. 2; xmv dno B. 14. nBQiBxofiBvov q>. 
16. FB] in ras. FV, BT p. hvxB Vp. 17. iaziv FV. 
19. jr] rj p. tBtQuymvmv q). 21. iattv FV. 22. 

nBQiBxofiivmv q), 23. tmv] add. m. 2 F. 24. Caov iativ V 
et p \iati), 26. taov iativ V9, p {iati). to] om. qj. 

26. iativ V. 27. tav] om.P. 28. iXaaaov F. iattv V. 
Post BA ras. unius fere lin. F. 29. B^] B-<4 qp. 



160 STODCEIiiN P'. H 

'Ev «pa rofs o^vymviois TQiytavoig ro ano r^g r^i^ 
dlEfav ymvCa.v vjtoznvovaris ixKBVQaq Tfrpayrovoi' ^Aor- 
rov iext t<Dv aao tmv rjjv o^stav yaviav TtBQiej^ovewv 
itlfVQmv T€tQttymva)v ttp nsQiExofidva Slg vno ts (uag 
3 tiov HEpi ti]v o^stav ycovCav, iqi' ijv ^ xa&szos nCntst, 
x«l r^S dTtolttfi^avofidvtjg ivTog vxo r^g xa&dTOV irpog 
T^ o^sCa yatvCa' otieq S6ei dEt^at. J^M 



Ta do^ivti EV&vyQttfifia tOov TstQaycovov 
10 ovST^^eaa&tti. 

"Eeza To do&iv sv9vyQttfi(iov to A' SeT Sij rp ji 
Evd^vyQttftfta [eov tsTQayavov avaf^aaa&ttt. 

HweeTttTm yitQ tS A EvQT>yQtt(i[ip teov naQttXXn- 
XoyQafiiiov 6g9oyoivtov to BjJ' eC ftiv ovv fffij iatlv 
15 71 BE rij EA, yEyovog av sttj ro imtax^iv. ffw- 
iattttai yaQ ta A EvQvyQdfifim taov tEZQdymvov tb 
B^- si d\ ov, ftitt Tmv BE, E^ fiECt,tav iotiv. SaT(o 
ftsi^BJV i} BE, xal ix^s^Xrja&a iTtl ro Z, xal xsCa&a 
rij E^ tarj 7} EZ, Xttl rETftijtffl-ra ij BZ dCxa Xttta 
20 ro H, xal xivtQm rto H, diaat-^fiKTt d\ svl Tmv HB, 
HZ ■^fiixvxliov ysyQd(p&oi ro B@Z, xal ix^E^kria^m 
71 /}E ini ro ®, xal inEt,Bvx^ta ii H&. 

'Ejtel ovv Evd^attt if BZ tizfiijtai slg y,s.v tea xattt 

1. Jv] inter e et v ma. 1 litt. V. S. iJLaaaov ?, S. 

ieziv V. 4. T«] om. F. S. ivTOs'] om. P. 11, lO fii» 

3o9fy p, 18. yHp] om. p. 14. B.4] BTJE p; io raa. V. 
IS. avvcaiatai] PBF, V tn. 2; ovwoiaroj V m. 1; fluv- 
laTarai p. 17. ov] poatea add. F. Post iiia 1 litt. (i7) 

eraa. F. 18. Upspx(a9-at. ip. 19. £Z] ZE BF. 20. ntii] 
postea add, F. Kt*re»] PB, F ra. 1; «^►Tpoi (iir Vp, F 

m, 2. HB] BK BF. ' 23. ovv] om, F. Seq. ras. 1 litt. 

V. BZ] in rae. 7, t^s] -s supra m. 1 V. 



ELEMENTOBUM LIBEB IL 161 

Ergo in triangulis acutiangulis quadmtum lateris 
sub acuto angulo subtendentis quadratis laterom acu- 
tum angulum comprehendentium minus est duplo rect- 
angulo comprehenso ab altero laterum acutum an- 
gulum comprehendentium, eo scilicet, in quod perpen- 
dicularis cadit^ et recta a perpendiculari ad angulum 
acutum intra abscisa; quod erat demonstrandum. 

XIV. 

Quadratum datae figurae rectiiineae aequale con- 
struere. 

Sit data figura rectilinea A. oportet igitur figurae 
rectilineae A aequale quadratum construere. 

construatur enim figurae rectilineae A aequale par- 
allelogrammum rectangulum B^d [I^ 45]. si igitar 
BE^ EA^ effectum erit^ quod propositum erai con- 
structum enim est quadratum BA datae figurae 
rectilineae A aequale. sin minus, alterutra rectarum 

BEy EA maior est. sit maior BE, 
et producatur ad Z^ et ponatur 
EZ = EJ, et 5Z in H in duas 
partes aequales secetur [I^ 10], et 
centro H radio autem alterutra 
rectarum HB, HZ semicirculus 
describatur B&Z^ et producatur z/E ad ®, et duca- 
tur HB. 

iam quoniam recta J3Z in partes aequales secta 




XIY. Simplic. in Arist. de coel. fol. 101 ; id. in phys. fol. 
12""; 14. Boetins p. 386, 23. 



Eaclides, edd. Heiberg et Menge. 11 



162 rroEXEiiiN p'. 1 

t6 H, lis di aviea xaia t6 E, t6 «po vTto tav BE, 
EZ 3rf(«fj;d^£vot' OQ&oydviov (itTU tov ajto Trjs EH 
TfTQa-ydvov ftfov ieTl rp «jro t^s HZ TiTpayrovp. fci; 
6i ^ HZ Tij H&- t6 «pa vxo rmv BE, EZ (leTa 
6 Tov ctno zijs HE teov ierl rra ajio t^s H&, ra de 
ttjro T^s H& tea jCTt tk flw6 Ttav &E, EH r£Tpa- 
yravK* t6 Rpa iw6 tcdv B£, EZ ft£T« rou arco H£ 
fffR ^St! Tofg «Jto Twv ®E, EH. xoivov a^p^fjQi^e&e} 
t6 iino T^g HE tttpdyovov kotnhv «pff to vno tav 

10 B£^, EZ 3rfpi£j;dftivoi' op&oyrartov fffov ^ffri tw ajii 
T^S £® TfTpayoJvM. ai-Xa to inh tav BE, EZ zb 
B^ ictiv' i'a-rj ywp ^ EZ t^ £^' t6 Kp« Bz/ srap- 
aA^ijAo^pR^ftov i'aav ietl tS uxo t^g &E iiTffo- 
ydva. teov di to B^ Tp ji iv9vyQtt[i(ia. xal t6 A 

n «pw fv^vypaftfiov fCov ^«tI tm ajro TTJg E& dvayQct' 
ipi^eofiiva) tiT^aymvq). 

Tra Squ do&dvti ei&vyQaiiiia ta A i'eov Tftpo-, 
yavov evviazaTai t6 «wo T^g £0 arKj-pKfpijffdjifvowl 
onfp ISei xoii\eai. 



1. 10] (tert.) Bupra m. J V. 2. EJf] H£ P. 3. 
— 5. H&] mg. m. 3 Vi iu teitu ras. tertiae partia lineae. 
fetCv if. i. fnco xuv BE, £Z] iirti xiv BE, EZ a^&oyoi- 

viov iD mg. traDBienam.lF, seq.ctSvBE, EZ tp^ xwv BE,EZ 
meiejoucvov ot&oymviov p. 5. HE] HE zitfByiavov p; 

Tiieavm.ou add. comp. m. 1 F. di ano] enan. F. 6. SaTiv 
Vip. Eif] Ppi HE BF, in caa. V, 7. EZ 5jie<eid(iMoi' 
c5e3'oyiai'ioi' p. HE] PBi rqe HE Vip, r^s E7f p. 8. i'an] 
loot ip. (ffiCf V. Toi-g] in ras. V. ©E, EH] Pp; 

(9E, H£ BF, V in ras.^ 9. HE] EH p. rav] supra m.2 
V. 10. nseiexo^iivov 6ij9oyiiviov] om. p. raiCf V, xaj 
id <p, 11, 10 B,dJ BFVp, Campanua; id vao zmv BE, EJ 
P. 12. £Z) ZE P. 13. iaziv V. 14. *ai] poatea add. 
comp. F; om, V, ,/| insert. m. 1 p, 15. iattv PV. 

amypttqiiiODfi/»'™] PBF; nVnypBipo/ifrni V, avayi/arpivTt p. 
" ' ^ F; Dfw^noTori Pp et V in raa, ttvayQnipit, 



6 

1 






ELEMENTORUM LIBER U. 163 

est in ff in inaequales autem in E, erunt 

BEXEZ + EH^ = HZ^ [prop. V]. 
sed HZ = HS. itaque BExEZ + HE^ = H@^. 
uerum BE^ + EH^ = HS^ [I, 47]. itaque 
B£:x JSZ + HE^ = dJS* + JSff». 
Bubtrahatur, quod commune est^ HE^. itaque 

BExEZ-=^E^. 
uerum BExEZ^ BJ\ nam EZ ~ EJ. itaque 
Bzi = ^£^ sed Bjd = A. itaque etiam figura rec- 
tilinea A quadrato^ quod in EB construi poterit^ ae- 
quale est. 

Ergo datae figurae rectilineae A aequale quadratum 
constructum est^ id quod in EB describi poterit; quod 
oportebat fieri 

p. 19. noiricai] dctfat FY. Ev%XBi6ov atoiX' § B, £v- 

Ttlsidov ctoixsimv xrjg Giatvos inSoceois fi F, riXog rov Ssxni^ 
Qov inoix^iov Tov EvnXeidov tov ystofiitQOv V. 



11 



L 



"Oqoi. 

a'. 'l6oi xvxloi eIgCv, cov kI Siufiszgoi ftfat elaivy,] 
i) cov at ix xmv xivtgmv Hatu. sleiv. 

P'. Ev9eta xvxXov i^dTtrse&ai XiyEzai, ^tigj 
6 anroiiivri tot xvxlov xal ix^aKko^tivtj ov tinvet iro»» 
KvxXov. 

y . KvxKoi itpdnre€& at dlX^^Xtav liyovrai 
orrtvEs axTOfiBVOi. aXX^^lcov ov ri^voveiv aXkijXovq. 
6'. ^Ev xvxXp raov aTtiiEiv «JTo xov xivzQov 
10 fv&etai. Xiyovrai, Zrav al dito rov xivrffov ia avrag 
xd&sroi ayofiivai tSai (00 vv. 

t'. Met^ov di aTtiy^tiv XiyBtai, i^' ijv r} (isl^wv 

xd&BTOS TCCXTEI.. 

ff'. Tji^fia xiixXov ietl ro %eqi,b%i (tevov ax^fta 
15 vico XB EV&sCas xal xvxKov ne^ipEpsCas. 

5'. Tjii^iittTog 6i yavCtt iotlv rj TisgiBxofiivr] vic6 
TE BV&sCtts xal xvxXov nsgKpEQBCas, 

ij'. 'Ev tfi^iiati S^ yiavCa istCv, otav ial ti}g 
negi^EtfsCas tov %ii^ftaros Xr}<p&TJ n efjftEtov xal iat 

Def. 1. Hero def- 117,3. BoetiuB p, 378,16. 2. Hero 

def. 116, 1. BoetiQB p. 378, 17. 3. Hero ib. Boetiua p. 378, 
19. 4—6. Hero def. 117,4. BoetiuB p. 379, 1. 6. Hero 

def. 38. BoetiuB p. 379, 5. 7. Boetiua p. 379, 9. 8. Hero 
def. 34. Boetina p. 379, 6. 



\ 






. PBFp; 



. PBFV. 



2. ilei 



J 






Definitio&es. 
I. Aequales circulj stint, quorum diametri aequalea 
nt, uel quorum radii aeqnalee. 
n. Becta circulum contingere dicitur, quaecunque 
cireulum tangens et producta nou secat circulum. 
III, Circuli inter ae eontingere dicuntur, quicunque 
bj&ter se tangentes non secant inter ae. 

iV. In eirculo rectae aequali spatio a centro distare 
' dicuntur, si rectae a centro ad eas perpendiculares 
ductae aequales sunt. 

V, Maiore autem spatio diatare ea dicitur, in quam 
maior perpendicularis cadit. 

VI. Segmentum circuli eat figura a recta aliqua et 
arcu circnli comprehensa. ') 

IVII. Segmenti autem angulua is est, qui a recta 
et arcu circuli compreLenditur. 
VIII. Angulua autem in segmento positus is est, 
boi siunpto in arcu segmenti puncto aliquo et ab eo 



I) Cfr. not. crit, ad p. 6, 1. 



8. tei] insert. ni. 1 P. taai.tl<iiv]ev. . .«iv intercedeute xas. 
JO Utt. F. 6. xiiivji V, sed corr. 6, Poat nvxXov odd, Inl 
fUjiixtfei titet P\ idem loco uocabuli oi Hero. Boetias, Cam- 
pfuins. 7. Ante %vkIoi rfts. 2 litL V, 9. awo] om. V, Hero. 
11. <sQt p. 12. t'] cum def. 4 CDuinuxit p. 14. laiiv Y. 
15. Post nrfiwtfifCas p mg. m. 1 pro acltolio add. q iiEi^ovci; 
ij /i»)ir«l»ov ^ /l«rTo»ios qfnxuJil^ii ; cfr. Hero. IB. ' ' ' ' " 



166 ETOrSEIiiN y'. 

avtov inl xa «tpara r^g sv^sias, ^ i6ti ^doig tQ< 
tfiijiitttos, iiti^Ev%9^aiv tv&tiat,, rj xiffiBiofiivvi yiovict- 
vjih tav ijti^tvx^^sioddv £v9Euav. 

fr'. "Orav 61 at nEQLi%ovaai t^v yfovCav ev&sl 
> aTtoXttfi^aviOSi riva TtiQLgyigetav , ix ixeivrjg Kiyi- 
rat (is^rixivai ij yavia. 

i'. Tofiivg Si xvxXov iStCv, orav Ttgog Tto xiv- 
tga rotJ xvxXov Ovffrafr^ yavCa, to weptejjo^svov Ox^f*" 
vzo ti rtoi' tiiv yavCav itE^tsxovaav tv&aiav xal r^s 
I ttJtoXafipavo^ivtjg vn avtav TtsQtfpt^siag. 

la'. "Ofioia tii^fiata xvxXav iatl ta Sexofii 
yavCas iffas, rj iv oXg aC yaviai feat aXXi]Xats sltsCv. 




Tov So&ivtos xixi,ov To xivtQov EiQeiv. 

"Eaxa o do&tls JciJxAos 6 ^BV- Set Sij tov ABF 
xvxkov To xivtQov tvQth'. 

^i-Yix^'^ rtg *^ii" ttvtov, ms hvxsv, tv&tttt ij AB, 
xttl rer^^o^frra Sixa xata t6 A arnttiov, xa\ aao Tot! 
^ xri AB Jipos oQ&ag TJx&at ^ AT xal Si^^x^at ijtl 
20 t6 E, xal ttxnrj69to r] FE Sixt^ xaxa xo 2. ' liyio, 
t6 Z xivXQOv iarl Toti ABF [xvxlov]. 

Mij yuQ, kAA' ei Svvazov, latat r6 H, xal ine- 
£ji5j;d(»0«v ttt HA, HA, HB. xal iTtsl Iffr] iatlv 
AA rfl AB, xoivi} Si ^ AH, Svo Si] aC A/3 
Svo rafg HA, AB fffai tielv sxateQa sxatiQa' xa\ 
^daig ri HA ^aast. Tfj HB ioxiv f«ij" ix xivxQov yaQ' 



1 



htl 

4 



Def, 0. BoetiuB p, 379, 10. 10. Hero def^ 36, Boefina. _ 

p. 379,13, II. Hero def, 118,2. Simplicius lii phys, f -■ ■ 
Boetiua p, 379, 16. I. ProcluB p. 302, 6. 



1. ^] PP; ?Tis BVp. ietiv BV. 




ELEMENTOBUM UBER m. 



167 



rectis ad terminos ductis rectae, quae basis est seg* 
meiiti, a rectis ductis comprehenditur. 

IX. Ubi uero rectae angulum comprehendentes 
arcum aliquem abscindunt; angulus in eo consistere 
dicitur. 

X Sector autem circuli est figura, quae angulo 
ad centrum circuli constructo a rectis angulum com- 
prehendentibus et arcu ab iis absciso continetur. 

XI. Similia segmenta circulorum sunt^ quae angu- 
los aequales capiunt; uel in quibus anguli aequales 
sunt [cfr. def.- 8]. 

I. 

Dati circuli centrum inuenire. 
Sit datus circulus ABF, oportet igitur circuli 
ABF centrum inuenire. 

producatur in eum utcunque recta 

ABy et in puncto A in duas partes 

aequales secetur; et a z/ ad rectam AB 

iperpendicularis ducatur /dF [I, 11], et 

producatur ad E^ et FE in duas partes 

aequales secetur in Z. dico, Z centrum 

esse circuli ABF. 

Ne sit enim, sed, si fieri potest, sit H, et ducan- 

tur HAj HAf HB, et quoniam A/l — AB, et AH 

communis est, duae rectae A^d^ ^H duabus ffz/, 

^B aequales sunt altera alteri. et HA = Jf5; nam 

V. Iw'] Iwt B. 7. dsj om. p. 11. xvxXa)y] PBp, Hero, 
Simplicius, Boetius; xvxAov Vqp. htCv V. 17. rntm P. 
19. Post AB ras. 1 litt. V. JF] FJ P. 21. xvxiovl 

om. P. 22. iniisvx^oiaav P. 23. xa^] om. 9. 26. dvoj 
dvaC Vp. ^ HJ, JB]JH, BJ F. 26. Carj katCv V. 
ydq] PB; yaq tov H FVp. 





168 ETOIXEiaN v'. 

yav^ afftt ^ v%o A ^ H ycovia tjj wtoo H^B tiSi] iozi 
ozttv Si sv&ita i7t' 8v9£tttv aza^^itOa raq i(psi,^s y*"- 
vitcg Itsae dlX^i.aig zoiij, opd^ ixaxiga zmv i^sav jo- 
vuSv ieztv 6(f&ii afitt iazlv tj vjco H^B. iezl Sh xal 

6 ^ VTto Z^B oq9"^' tetj Kpa 71 vno Z^B t^ vxb 
H^iB, ij (*£t§rav xjj iXttzzovf owsp iezlv aSvvatov. 
oint «p« To H xivtfov iatl tow ABF xvxXov. ofioiae 
St] tff(|o(t£v, OTt, ovS^ aXJ.0 ti nXijv to 

To Z apa erjiiilov xivttfov iotl zov ABF [xv- 

10 xXov\ 

Hoffiefia. 
'Ex J^ tovzov ipttvBgov, oit iav iv xvxla sv&eta 
rtg tv&tiav tivtt Sixa xkI nrpog op&ag t^jivti, ixl tiji 
TtftvotJffjjs ietl t6 X8VZQ0V zov xvxXov. -— ontQ idti 

15 Xoi^Cttl. 

'Eav xvxXov ixl n^s ntpnpefidag Xrj^&fj Svi 
xv%6vttt erjfista, tj inl ta erifieta iai^Bvyvviidvi 
tv&ata ivtog ntetlxai tow xvxXov. 
20 "Eeta xvxXos b ABF, xal inl z^g «iQi^efftittg 
ttirtov tiXijrpQ-a Svo xvy_6vta eTjfiBta za A, B' Xiya, 
ozt 7j «Jto Tov A iml x6 B im^svyvvfiivTi £v9tttt 
Tog xtetlxtti xov xvxXov. 

Mi] yap, aXX' ti Swax6v, ntnzixa ixzos a>q ^ 
26 AEB, xal tlX^^ip&a z6 xivtQov zov ABF xvxXov, xal. 



Ptop. I «df. Proclns p, 804 6. SimpliciuH in phya. fol. 14',; 



I. imv foij p. 3, opft)] ifltw p- iacov] om. P. 
;«■] om. p, WiJBl JHB ip. 6. H^S] in ms. 1 
!ii<D« iH M^tooi P. 7- lariv V. ABF] HSr f (non 

8.' ovS"\ ovSi P. 9. oen] om. F, hj(v PV. 

tXov] om. P. H, KopiofiK] om. F. 12, tig tv&tia V. 



t 

J 

I 



ELEMENTORTJM LIBER m. 169 

radii sunt. itaque L^^H ^ H^B [1^8]. ubi uero 
recta super rectam erecta angulos deiuceps positos inter 
se ae^piales ef&cit^ uterque angulus aequalis rectus est 
(1 def. 10]. itaque LHAB rectus esi sed etiam L Z^B 
reetus esi itaque L2AB ^^ H^B maior miuori; quod 
fieri non potest. quare H centrum non eet circuli 
^BF. similiter demonstrabimus ne aliud quidem uUum 
punetum eentrum esse {meter Z. 

Ergo Z punctum cemtrum est circuli ABF. 

Corollarium. 

Hi^ manifestum est^ si in circulo recta aUqua 
atiam rectam in duas partes aequales et ad angulos 
rectos secety centrum circuli in recta secanti esse. ^) — 
quod oportebat fieri. 

II. 

Si in ambitu circuli duo quaelibet puncta sumpta 
erunt^ recta pimcta coniungens intra circulum cadet. 

Sit circulus ABF^ et in ambitu eius duo quaelibet 
puncta sumantur A^ B. dico^ rectam ab ^ ad £ duc- 
tam intira circulum casuram esse. . 

Ne cadat enim^ sed^ si fieri potest, cadat estra ut 



1) Nam mTJ in media^JB perpendiculari erecta centnjun 
erat positmn; ceterum hoc corollarium quasi parenthetice poni- 
tur, ita ut uerba oneQ iSsi noiriaat lin. 14 ad ipsum problema 
I referuntur; cfr. III, 16, al. 

14* i^tiv V. noi^aat] Saiiai P. onsQ idst notr^ccu] om. 

p. 18. criiLBia xv%6vta p. ra] PBp, V m. 1; xa avxd P, 
V m. 2. 



r 



no 



2T0IXEIJ2N v'. 




'Ead ovv H0I] iarlv 17 ^A ry JB, [611 aQtt xal 
ytovia rj vab ^AE ry v^o ^BE' xa\ iml tQiydvov 

5 TOV ^AE (lia sEXEupa ffpoOex^c^Aijrat ?j AEB, iiEC^mv 
aga ^ vao ^EB ycovCa ti^q vitb /3 AE. torj S% ij imh 
A AE T^ vTth ABE- (lEi^mv apa i} vxo ^EB t^g 
lOTO ABE. vxo S^ T^v ^figovK yaviav ij (lEi^av TClsvQa 
vnoTsCvEf (iEi%<av aga ^ ^B rijg j^E, ("ffjj Se ij /JB 

Tfj ^Z. ii-Ei%03v a(fa ^ AZ Ttjg AE ij ilatTctv r^g 
(lEi^ovog ■ oTtEp iatlv aSvvuTov. ovx apK ij aab vov 
A inl To B im^Evyvvfiivi] tv&Eia ixTog aeaetTai tov 
xvxkov. o(ioitas Sij Sei^ofisv, ori ovSh in' avr^g rq^fl 
nEQKpEffBiag' ivTog aQa. 

B 'Eitv apa xvxXov inl r% TTEQtg^tQeCas XiJ<p9y Siuili 
Tvxovia atjfieta, !] ial tu ajjfiECa E7titEvyvvii.Evr} Ev&stn 
ivrbg nEaBttai, roi? xvxkov owip iSei Sefiai. 

y'- 

'Ettv iv xvxi.a eii^ElK ti^ Sta tqv xivTQOvM 
fi Bv&etav xtva (tij Sta rov xevtqov Si^a TEiivtj, 
xa\ Kpog oQ&ag avri]v refivei' xal idv npoff | 
OQ^ag avt^v tjfti/jj, xal SC^a avtijv rifivEi. 

"Ecto xvxkoi 6 ABV, xal iv avra ev^etd Ttj Sia 
roi xivTQOv ij FjJ £v9£tdv riva fii] St-a Toi) xivTQov 



l. .JJ] JJ V, 3. JZEJ PBp; V m. 1; JZ litl 10 S 

y m. 3; iit F poat JZ etas, E et ^;e1 id supra ecr. m. S, 
S, is«l ovr] m>) Js«/ P. 4. ^ y<orla ^ P. Tpi/fecoo] in iwi, 
comp. m. 3 r. S. .JE6] PB, p (r ^ id ras.); EB mipn 

a<:i. J m. t F; AB ini xi B V e corr, 10. tnj tqe F. 

apa Ks/ p. 13. jq] gorr. ex 9^ m. S V. 11. Sfti mtoti- 

rei P. 15. Kvxlov «fo p. 16. vij^cta Tvzorra p. h] 



I 




ELEMENTORUM LIBER m. 171 

AEB, et sumatur centrum circuli ABF [prop. I\, et 
sit A, et ducantur AA, AB, et producatur AZE, 

iam quoniam AA = AB, erit 
LAAE^ ABE [1,5]. 
et quoniam in triangulo AAE unum 
latus productum est AEB, erit 

L AEB > AAE [1, 16]. 
uerum 

itaque L AEB> ABE. sub maiore autem angulo 
maius latus subtendit [1, 19]. itaque AB > AE. sed 
^B aa AZ. itaque AZ > AE minus maiore; quod 
fieri non potest. ergo recta ab ^ ad J3 ducta extra 
circulum non cadet iam similiter demonstrabimus, 
ne in ipsum quidem ambitum eam cadere; intra igitur 
cadet 

Ergo si in ambitu circuli duo quaelibet puncta 
sumpta erunt, recta puncta coniungens intra circulum 
cadet; quod erat demonstrandum. 

m. 

Si in circulo recta aliqua per centrum ducta aliam 
rectam non per centrum ductam in duas partes ae- 
quales secat, etiam ad rectos angulos eam secai et 
si ad rectos angulos eam secat, etiam in duas partes 
aequales secat. 

Sit circulus ABF, et in eo recta aliqua per cen- 
trum ducta F^ aliam rectam non per centrum ductam 

xa avxd tp (in mg. transit), V m. 2. 17. dsi^ai] supra add. 
noifiaai F m. 1. 21. zifivsi} P, ts(i6i BFVp; sed cfr. 

p. 174, 19. 22. Tsnvsi] P; TS(i.si: BFVp. 



172 ETOrXEIiiN y'. 



1 



Ti}v AR SC%a Tt[iveTai xara t6 Z eij(t£ioV kiyei, 
xa\ itQoq o^Qas avzriv zifivn. 

EiXrjfp&tD yaQ ro xivTQOv rov ABV xvxXov 
ieta t6 £, xal ^xst^vx&iaaav ai BA, EB. 
6 Kal iitsl l^eri iszlv tj AZ ffj ZB, xoii'^ de rj ZE, 
Svo Svelv CCtti [tiaiv] ■ xal ^aeis ^ Ej4 ^aeti xy ES 
le^ri' ymvia aga ^ iixo AZE yavia ty wto BZE iot] 
ieriv. OTav Ss sv&eta i% Ev&Eiav eTa&iiaa tag itp- 
^ivs yaviag teag aAiijAKis itoijji, opS^ ixatspa zmv 

10 temv ycavmv ieriv ixari^a ap« rav uwo AZE, BZE 
OQ&T} iCTiv. )] FjA apa Sia tov x^i^pou ovea t^w 
jiB fiT} ditt Tov xivTifov oveav Sixtt TifivovOa jmI 
jtQoe OQ^tts rifivBi. '•■ 

'AXXa S^ i} r^ jipfAB Xf^g dpfraff tefiviTa' X^yn^^ 

16 oTt xal Si^a avTrjv rifivH, TOVtieTiv, ori tarj ierlv ^ 
AZ rri ZB. 

T&v yttQ avrmv xaTaexEvae&ivtaiv, ijtil fffjj iarlv 
ij EA rfi EB, terj ierl xal yavia i) iao EAZ t^ 
vno EBZ. ierl Sl xal dpfl'^ ^ vxo AZE oqQ^ rfj 

20 lOTO BZE terj- Svo aga TQiytava ieri ra EAZ, EZB 
TKg Svo ymvias Svel yaviais teag i%ovTtt xal ftiav 
xIevquv (ii^ TtXevQtt Cetjv xoiv^v avtcav t^ EZ vao- 
Ttivoveav vno (liav rmv Cecov ymvimv xal vas ioinas 
ttQtt nXiVQas rats kointtis nXsvQatq tOag «|«t ' teij «Qtt 

2& rj AZ TTJ ZB. 



2. TffieC F. 6. ZB] corr. ei BZ m. 2 V; BZ B. 

ivo 3n BVp, io B seq. »-X— ;< p^i"'*] om. P; ehC p. 

EA] AE V. 7- SZE] BZB P. 9. S^iiJattv^Bp. 

10. ^ITiv] om. Bp; HDpra comp. m.S V. 10. 6e9fi ai)a larlw 
inareQa imv vno AZE, BZE P. AZE, BZE] io tsa. F. 

11. leuv] comp. BTipra fcr. F. Fid] F poBtea inaert. V, 
13. ari^v Tffivtt V, H. iq ^ai V. TJ] T postea ineert 



1 



ELEMBNTOBIIM LIBER m 



173 



jiB in duas partes aequalea secet in puncto Z. dico, 
eandem eam ad rectos angulos secare. 

sumatnr enim centrum circoii ABV [prop. I], et 
sit E, et ducantur EA, EB. 

et quoniam j4Z = ZB, conimunis autem est ZE, 
duae rectae duabua aequales sunt. et EA = EB. ita- 
que LAZE =^ BZE [1,8]. ubi uero recta auper rec- 
tam erecba angulos deinceps poaitos inter se aequales 
efflcit, uterque angulus aequalis rectus est [I def. 10]. 
itaque uterque angulus AZE, BZE rectus est. ergo 
Vjd per centnim ducta rectam AB non per centrum 
dactam in duaa partes aequales secana eadem ad rec- 
tos angulos aecat. 

Uerumr^rectam.,iB8d rectos angu- 
loBsecet. dico,eandemeam in duaspar- 
tea aequales secare, h.e.ease AZ = ZB, 
uam iisdem comparatis quouiam EA 
= EB, erit etiam L EAZ = EBZ 
[1,5]. uerum etiam L-^ZE-^BZE, 
sunt. itaque'} duo trianguli aunt EAZ, 
EZB duos anguloa duobus aequales habentea et unum 
latus uni lateri aequate EZ, quod commune est eorum, 

(b altero angulorum aequalium aubtendens. itaque 
iam reliqua latera reliquis lateribus aequalia habe- 
int [1,26]. ergo AZ=ZB. 
i 
18, ^. «(«pou mg. V (Bcliol.), lativV. 19. EBZl 
litt, BZ in ros. V; corr. ei EZB F. iaviv V. 20. Kpo] 

om. PBF; comp, sapra scr, V m. 3, t^lyana] -ymvu erka. 

B. imv V. 




1) Cnm SfB lin, EO in omnibua bonia coiiicibua omiasum 
i, fortaaae potins pro taii iaxl *ai lin, 16 scribendum: fa<j 8i 



174 rroiXEmN y'. 

'Eav «pa iv xvxla ev&cta ttg fiwi tov itdvtQotf 
tv&iZav TLva [lij Sia %ov xivxQov dixa ziftvy, xal 
;tpos og&a^ avrijv zifivei' xal iav apos op#KS «rr^v 
Tiiivf], xal d^x^ Kvriiv tifivfi' owfp Mei St^^ai 



VXQOV^ 

*al 

I 






'Eav iv xvxkc] Svo tv&etai ti^vtaoiv aXk^ 
lag fi^ Sia Tov xivTQOv oveai, ov ti(ivoveiv 
aXXr^Xag S{%a. 

"Eetta xvxXo^ 6 ABFJ, xal iv avra Svo Ev^stai, 

10 ai jiV, B^ Ttfivitaeav aXX^^Xag xaza xo E [lij dia 

roiJ xivtQov oveaf Xiya, ort ov tifivoveiv ciXX^Xas 

""" 

Ei yap Swatov, teyivitaeav «XXrjXaq 'Sl%a cSfftrf 
ftfjjv slvat T^v fiiv AB rij EF, t^v Sh BE *g EA\ 

15 xaX siXritfi&a xo xivtpov tov ABF^ xvxXov, xal li 
zb Z, xal iaetBvx&ca 17 ZE. 

EtcsI ovv ev&eia ttg Sia rot' xivt^ov tj ZE 
&ei:av Tiva (tij Sia tov x^vrpor r^v AF Sl%a tifivei, 
xal TTpbg OQ&ag avzijv tifivei " opO^ ap« icrlv 17 vxb 

20 ZEA' xaXiv, inel evQeta riq ii ZE ev&eiav Tiva t^v 
BjJ SCxtt Tifivei, xal itQog op9as avziiv ti^vBi.' opfr^ 
«pa ri V7tb ZEB. iSe{%9ri Sl xai 7} vxb 7.EA opftij' 
iVi] upa 17 v%b ZEA tr^ vno ZEB ^ iXaTtmv zij fieSi 
Sovf oiteg iezlv aSvvatov. ovx apa al AF, B^d tii 

26 vovOiv aXX^Xag SCxa. 

l._ h «UKlai] om. p; b«k1o> comp. V, Iv 
fv&eldv xaia — 1. Wfivei] Kal xa i^ifs PBY, 
refiiTi] xal rc /J^s F. 4. Tffitij] -fn-ji in n 
lifior P. 18. B^ yap — 14. »0 ^^] ■" '^ 

rfflj* p. 18. ft^ JiO TOr MEIITPOU] Pp: 

pvfi] PBpg); itufiV. ;«^ P. aO, 





ELEMENTORUM LIBER m. 175 

Ergo si in circulo recta aliqua per centrum ducta 
aliam rectam non per centrum ductam in duas partes 
aequales secat^ etiam ad rectos angulos eam secat; et 
si ad rectos angulos eam secat, etiam in duas partes 
aequales secat; quod erat demonstrandum. 

IV. 

8i in circulo duae rectae inter se secant non per 
centrum ductae^ in duas partes aequales inter se non 

secant. 

Sit circulus ABF^ et in eo duae 
rectae Ar, Bd non per centrum 
ductae inter se secent in E. dico^ 
eas in duas partes aequales inter se 
non secare. 
nam si fieri potest, in duas partes aequales inter 
se secent, ita ut sit AE — EF et BE = E^, et su- 
matur centrum circuli ABFjd [prop. I], et sit Z, et 
ducatur ZE. iam quoniam. recta per centrum ducta 
ZE aliam rectam non per centrum ductam AF in 
duas partes aequales secat, etiam ad rectos angulos 
eam secat [prop. III]. itaque L ZEA rectus est rur- 
sus quoniam recta ZE aliam rectam Bjd in duas 
partes aequales secat, etiam ad rectos angulos eam 
secat [id.]. itaque L ZEB rectus est. sed demonstra- 
tum est, etiam i ZEA rectum esse. quare 

LZEA^ZEB, 
minor maiori; quod fieri non potest. itaque rectae 
AF, Bd in duas partes aequales inter se non secant. 

V; in' F, corr. m. 2; om. B. 21. BJfi^ dui tov liivtQov 

F, V m. 2. tinvsi] (alt.) PBVp; ts[ist F. 23. iXdoaoiv 

F. 24. iath] PBp; om. Yq>. 



176 ETOIXEIiiN v'. 1 

'Eav ttQa iv xvxl^ 6vo svQ^ittti zifivtoSiv (^AAijAag 
fii] Sia Toi/ xivTQov ovotu, ov riybvovaiv aKi.iiKtts Si%i^' 



5 'Eiiv 8vo xvxkoi, TS(ivmai.v alX^^ovg, ovx 
lerai, «vrrov vh auro xivtQov. 

/Jvo ya^ xvxi.oL ot ABF, r^H zsnvitmaav aX- 
Ajjioug xuTu za B, r <tri(tBta. Xeyio, oii ovx Sazat av- 
tSv t6 outo xivTQOV. 

10 Ei yap dvvavov, ^azio to E, xal iwElsvx&a rj EF, 
xal Si7]x&(0 Ti EZB, ths ^tvxev, xal izsl ro E Otj- 
listov xivtQov ietl rotJ ABV xvxkov, fffTj ietXv ^ 
EF r^ EZ. naKiv, insl to E atifisiov xivtQov ietl 
Tou r.JH xvxlov, rarj iatlv ^ EF tfj EH- iSsCx&ri 

16 S\ ri Er xal ty EZ tat}- xal ij EZ a^a t^ EH 
ietiv ftfjj 7} ilaaeav t^ ftsi^ovf oirep iatlv dSvvaTov, 
ovx UQa To E Orjjistov xivtQov ietl tinv ABF, F^H 
xvxlav. 

'Eav uQtt Svo xvxloi. tinvGiaiv allTJXovs, ovx Setiv 

BO avtav xo avto xivxQov okeq iSsi Sst^ai. 



'Eav Svo xvxXoi i<paittaivTai, dXk-qXmv, ov» 
Settti. avt&v to avto xivtgov. 



■ 



2. fnj 6ia — 3lza] xsl ta fj^s BFV. 7, rjH] JH 

V. 8. B, r] r, B p. 10. ET] TE p. 11. ftuie p. 
13, iiSTiv V. T0«] 1)18 P. 18. iativ V. 14. ET] TE 

P, 15. PoBt df ! litt. eras. V. EZ] <alt.) ZB P. 16. 

ftnj ^ac/* p. iXattmv BVp. ^oii'v] om. V. 17, 

V. 19. laiaiVp. 22. dliniiov Ivzos V et P m. i 




ELEMENTORUM LIBER m. 177 

Ergo si in circulo duae rectae inter se secant non 
per centrum ductae^ in duas partes aequales inter se 
non secant; quod erat demonstrandum. 

V. 

Si duo circuli inter se secant, non habebunt idem 

centrum. 

nam duo circuli ABF^ F/IH 
inter se secent in punctis B^F. dico^ 
eos idem centrum habituros non 
esse. 

nam si fieri potest^ sit £, et ducatur EF^ et edu- 
catur EZH utcunque. et quoniam E punctum cen- 
trum est circuli ABF, erit Er=EZ, rursus quo- 
niam punctum E centrum est circuli F^H, erit 
EF = JEH. sed demonstratum est etiam EF = EZ. 
itaque etiam EZ = EH, minor maiori; quod fieri 
non potest. itaque punctum E centrum circulorum 
ABFj rJH non est. 

Ergo si duo circuli inter se secant, non habebunt 
idem centrum; quod erat demonstrandum. 

VI. 

Si duo circuli inter se contingunt, non habebunt 
idem centrum.^ 



1) Euclides eum casum, quo circuli intra contin^untf 
ut obscuriorem sibi demonstrandum sumpsit; nam ubi circuli 
extrinsecus s&^contingunt, propositio per se patet. ceterum 
demonstratio Euclidis de hoc quoque casu ualet. quare ivtog 
lin. 22 mera interpolatio est, ut etiam e codicum ratione ad- 
paret (om. Gampanus). 

Enclidea, edd. Heibeig et Menge. - 12 



178 



ETOISEIBN y\ 




Jvo yuQ xvxXoi ol ABF, F^B iqtaardffQ^aaav 
AijAcjw xaru tb F e^fiEiov' Xiyei, oTt ovx ieiai 

t6 aVTO xivTQQV. 

Ei yag tftwatof, ierto to Z, xal ijiB^8vx&tt> 17 ZI^' 
6 xal tft^^f^&to, roj Stv%ev, 17 ZEB. 

'Eitel ovv To Z erj[ttlov xivTffov ierl tow ABP 
XVX2.0V, t0rj iCTiv i] ZF Tij ZB. naXiv, intl to Z «1/- 
ftfrov xivTQOv iarl row F-Jfi xiJxAot/, fffi; ^oriv ^ ZF 
T^ ZB. ^dEtz*)? 6b 17 Zr T^ ZB aet]- xul 7) ZE a(fa 
10 r^ ZB ietiv tffri, i] ikaTXiov tjJ fi,Eit,ovf ortsQ ierXv 
aSvvatov. ovx aQa t6 Z e^fitlov xivTQOv ievl Tmv 
ABr. r^E xvxkav. 

'Euv aQa Svo xvxkoi ifpaitTavTut akk-qlav, oiJw' 
Satai avTwv t6 o;wr6 xivtQov oacQ iSii. Stliat. 



^Eicv xvxkov iTcl T^s SiafiiTQov kTi<p&^ ti ojj-- 
(i£tov, o [i-^ iffTi xEVTpoi' Tov xvxkov, ano Si 
Toil arjiitiov ntQog tov xvxkov aQoffxiitTiaeiv 
sv&BtaC jTivfs, lisyietti [ihv ietai, iip' ijg ro 

xivTQov, ika%Cazri Sh rj /oiffij, Tav Si akkoiv 
atl r} iyyiov r^g dia roi; xsvtqov t^g ajliDTEQOV 
(itC^av iffTlv, Svo Sl [lovov Ceui ujto row oij- 
fiEt^oti nQoajteeovvtai TtQog tov xvxkov itp' ixd- 
tsQa tijg ikaxCetrig. 

i "EffTo) xvxkoq ABV/i, StafieTQog 61 avtov Savn 
17 A^, xal ixl T^g A^ tiX^fp&co ti a7}(iEtov t6 Z, 
ftij iffTt xivtQov rov xvxkov, xivXQOv Si tou xvxj 

1. i:tctf<i9ceiiav P et F m. I (corr. m. 2). 2. letiii.'] iati* 
Vp, 6. lnTiv V. 7. ZB] &Z P. «ol« — 8, r^dJE] in 
«w. p. 8. iniv V, 9. ai «u/ p et F m, 3, 10, lU.9- 



1 



exta _ 




ELEMENTORUM LIBER m. 179 

nam duo circuli udBF, F^E in puncto F inter se 

contingani dicO; eos idem centrum habituros non esse. 

nam si fieri potest^ sit Z^ et ducatur Zr, et edu- 

catur ZEB utcunque. iam quoniam punctum Z cen- 

trum est circuli ABF, erit Zr=ZB. 
rursus quoniam punctum Z centrum 
est circuli FJE, erit Zr=ZE. sed 
demonstratum est Zr= ZB. quare 
etiam ZE= ZB minor maiori; quod 
fieri non potest. itaque Z punctum cen- 
trum circulorum ABF, FjdE non est. 
Ergo si duo circuli inter se contingunt; non habe- 
bunt idem centrum; quod erat demonstrandum. 

VII. 

Si in diametro circuli punctum aliquod sumitur, 
quod centrum circuli non est; et ab hoc puncto ad 
circulum rectae aliquot adcidunt^ maxima erit ea^ in 
qua est centrum^ minima autem reliqua, ceterarum 
autem proxima quaeque ei, quae per centrum ducta 
est, remotiore maior est^ et duae solae aequales ad 
circulum adcident a puncto illo in utraque parte mi- 
nimae. 

sit circulus ABF^, diametrus autem eius sit Ajd, 
et in AA sumatur punctum aliquod Z, quod non est 
centrum circuli, centrum autem circuli sit E, et o, Z 

(fav Fp. iorh] om. p. 11. iatlv V. 13. iqxxntoavTai] 
i(p- add. in. 2 F. aXXiJXcoi; ivtog V. 17. iativFY. 

19. tivsg^ mv (ila (ihv dia tov tiivtqov ai 8^ Xomal (og itvxsv 
F. 20. dl 17] supra m. 2 F. di] d' FV p. 21. iyysvov P. 
dnmtSQa} P. 22. iati PBp. svd^siai taai Bp, V m. 2. 
Tov avtov BVp. 25. 6] postea add. V. 6i] om. p. ^at(o] 
om. p. 27. iativ F. %ivtQov] (pr.) in ras. p. Ss] insert. p. 

12* 



r 



180 STOIXEIiiN y . ^H 

ftfTOj xh E, xal ano Toti Z jrpog rov ABF^ xvttkov 
nQoaiciJCT{Tm(}ttv tv&ital Tivtg aC ZB, ZF, ZH' liya, 
OTi (uyiSTi) (i^v ietiv ^ ZA, iktt%(0xri Ss i\ Z/i, 
Tmv di aXlav r} (liv ZB t% ZF iiEi^mv, 57 dh ZF 
5 Tijs ZH. 

'Enc^wxQ-toeav yaQ aC BE, FE, HE. xttl iatl 
navtog T^iyavov at Svo nlevQal r^g loiaijg [isi^ovig 
litSiv, at aQU EB, EZ ttjs BZ fitiiovdg eIOiv. fei; 
S^h 71 AB T^ BE [ttt a(ftt BE, EZ teai del TyAZ\- 

10 nii^av ttQtt f] AZ zijg BZ. ndliv, intl lei\ iezlv ^ 
BE x^ rS, xoivii dh ri ZE, 6vo S^ at BE, EZ Svel 
ttttis TE, EZieai sieiv. aXXa xal yavia ^ vao BEZ 
ytoviag T^g vzo VEZ (t.Bii,a>v' ^aetg UQtt ij BZ ^a- 
esaig r% VZ fiei^av ietiv. Sia tu avta di] xal ^ 

16 rz T^s ZH ftii^av ietiv. 

ITtthv, insl at HZ, ZE t% EH iiEi^ovis e^tfw^ 
fffij di ^ EH tfi EJ, at KQtt HZ, ZE t^s Ed ^«^^ 
^ovig Eieiv. xoivi} ttg>^Q^e9to ^ EZ' lomii aQtt rj HZ 
Aoirt^g rijg Z^ fifi^wv ietiv. iisyieTij filv apor ]j ZA, 

20 iXaxietj] Si ij Z^, fiBi^tav Si 17 (liv ZB tijg ZF, ij 
3i Zr T^s ZH. 

Aiya, ott xal axo tov Z efi(t£iov Svo fiovov ''Attfl 
TtQoexieovvtai jrpog toi' ABF^ xvxkov itp' ixatBpa 4 
T^S Zjd iXttxiertjg. awtetdtiD yuQ TtQog rij EZ sv~ 

25 Qsitt xaX rp Kpog ttvti] erjfieia ta E rg vjto HEZ 
yavitf Cam ^ irjto ZE&, xal im^evx&ta ^ Z&. ixeX 

1. xilxiou qo. 3, lativ\ om. FV. ZA] 9 (eraa. Z.d). 
4. Zr] coiT. m. 3 es Hr V; TZ P. Zr] FZ F et m.2 
V. 6. IH V- 8. lioi*, ('(Jij df ^ AE tp BE. ai a^a BE 

F. ar EB, EZ afa P. T^e BZ — 9. EZ] om. F. ft 

v( E] iij uaa. m. 2 V. ai aea — AZ] mg. va. 2 P. titlr 

B. 10. Aute BZ ras. 1 litt. V. 11. Se] ova. PB. imti] 




ELEMENTORUM LIBER m. 181 

ad circolam ABFjd adcidant rectae aliquot ZB, ZF, 
ZH. dicOy maximam esse ZA, minimam autem Zjd, 
ceterarum autem esse ZB > ZF et Zr> ZH. 

ducantur enim BE, FE, HE. 
et quoniam cuiusuis trianguli duo 
latera reliquo maiora sunt [I, 20]^ 
erunt EB + EZ> BZ. sed 
l^ AE = BE. 

quare AZ> BZ. rursus quoniam 

BE = FE, communis autem Z E, 

duae rectae BE, EZ duabus FE, 

EZ aequales sunt. uerum etiam, L BEZ > FEZ. 

itaque BZ> FZ [l, 24]. eadem de causa etiam 

rz > ZH. 

rursus qupniam HZ + ZE> EH [I, 20], et 

EH=EJ, 
erunt HZ + ZE > EA. subtrahatur, quae communis 
est, EZ. itaque HZ> ZA.^) itaque ZA maxima 
est, ZA autem minima, et ZB > ZF, Zr> ZH. 

dico etiam, duas solas aequales a puncto Z ad 
circulum ABFA adcidere in utraque parte rectae 
minimae ZA. construatur enim ad rectam EZ et 
punctum eius E angulo HEZ aequalis L ZE@ [I, 23], 

1) Hoc Enclides ita demonstrauit: 

HZ+ZE = EJ + x. 
£Z = EZ. ergo HZ = ZJ + x [x. ^vv. 3], h. e. HZ>ZJ, 



dvo PV. 14. iar^v] PBF; comp. p; iari V. 16. Zif] HZ 
P. icxiv] PFp; iati BV. 18. datv] PF; stai BVp. 
19. loin^ ^ P- Z^] supra m. 1 V. iativ] PF; iat£ BVp. 
ILiv] supra m. 1 F. 20. tmv d' aXXa^v fisi^aiv filv rj ZB 

p. 21. T^ff] Tw V. 22. taai] PF; sv^sicii, taai BVp. 

28. ABFJ] J add. m. 2 V. 24. ZJ] om. p. 




ovv fffTj Efftfv rj HE TJi E&, xoivij Si ij 
Sii aC HE, EZ dval rafg ®E, EZ iaat siaiv xal 
ymviK ii vnh HEZ ymvia r^ vxo @EZ fffij" fidaig 
aga ^ ZH ^aaat TJj Z@ tei\' iatCv, Xiya tfij, "n vfj 
fi ZH aAlii tai] Qv XQoaitiaBlxai utffos tov xvxXov dno 
rov Z atifitiov. bI yag Svvatov, m^ogmazitm ii ZK. 
lal iTCsl ij ZK ty ZH Harj iaziv, uXXa tj Z@ t^ ZH 
[ftfjj ietiv^ xui 71 ZK aQ« tjJ Z® iariv Ferj, ^ Syytav 
j Sta roir xeWpou tt] axtazEQOv HOr)' oneQ aSvvatov. 
10 ovx aQa axo rov Z atjfiilov itEQct xig TtQositEasttett 
jrpog TOi/ xvxlov tari t^ HZ ' fiia ap« fiovi;. 

'Eav apa xvxXov ial tijg StanitQov IrjpS^ xt <jij- 
(isiov, imj iett xivxQov tov xvxlov, aito Sh toi) Oifi 
fieiov Jtpog ToK xvxkov nQosnintmatv Ev&tlai ttvssi 
16 (leyieti] fiiv latat, i<p' ije to xivtQov, iXa%ietyi Ji i} 
Aotm;, ttoi' tfJ aXlmv ael ij iyyiov t^g dt« rou xiv- 
tQOv t^g anatcQov (t£it,mv iativ, Svo Sh fiovov Ceat 
ano tov ttvrov aijfisiov TtQoeaeeoiJvTttt xqos tov xv--' 
xXov iip' sxciteQa t^g iXaxiat^g' oxsq SSst Sei^ai. 



80 



'Eav xvxkov Ai/qoftjj rt aijftffov ixtos, «! 

8i rov erjnsiov tiqos tov kuxAov Staj^^mOiv 

ev&eitti ttveg, o>v (lia (liv dtu rou xivtQov, at 

Ss koinai, mg itvxev, rav (thv TtQog tijv xoHriv 

26 neQnpeQSiav irpoOsrtJtrouOwv Ev9sttov (isyiat^ 

2. HEl EH F. £^ai'>'] PBF; eiai Vp. 4, ^oii* rn; 

p. iativ] latl V. afl om. V (yciD add. m, 2), Si F. 

5. ZHl H eraa. V. G. ^] ras ^ BPp. 7. ih ZXl e 

corr. m. I V. ietiv tm, Pp. oW<i] dXl' BPj riUi pijr 

»0^ P. ZH] corr. ex ZE V m. 1. 8. /'aij iativ] om- P; 

f(t»j F; lativ lOTi Vp. os«] om. F. ZS] SZ P, rai] 



v 

i 

i 



I 



ELEMENTORUM LIBER IIL 183 

et ducatur ZS. iam quoniam HE = ES, et EZ com- 
munis est^ duae rectae HE, EZ duabus SE, EZ ae- 
quales sunt. et L HEZ = 9EZ. itaque ZH= ZO. 
dico igitur^ nullam aliam rectae ZH aequalem a 
puncto Z ad circulum adcidere. si enim fieri potest, 
adcidat ZK. et quoniam ZK = ZH et Z& = ZH, 
erit etiam ZK = Z&^ propior remotiori; quod fieri 
non potest [u. supra]. itaque a puncto Z nuUa alia 
rectae HZ aequalis ad circulum adcidet. ergo una sola. 
Ergo si in diametro circuli punctum aliquod su> 
mitur^ quod centrum circuli non est, et ab hoc puncto 
ad circulum rectae aliquot adcidunt^ maxima erit ea, 
in qua est centrum^ minima autem reliqua^ ceterarum 
autem proxima quaeque ei^ quae per centrum ducta 
est^ remotiore maior est^ et duae solae aequales ad 
circulum adcident a puncto illo in utraque parte mi- 
nimae; quod erat demonstrandum. 

vin. 

Si extra circulum punctum aliquod sumitur^ et ab 
hoc puncto ad circulum rectae aliquot educuntur; qua- 
rum una per centrum^ ceterae autem utcunque duc- 
tae sunt; earum rectarum, quae ad cauam partem am- 



VIII. Eutocius in ApoUon. p. 12. 



ioth V. 17] om. F. ^YYSiov P. 9. t^I r^g PBVcp. 
ftfi}] del. August. ddvvatov] hic seq. demonstratio alia, quam 
in app. recepi. 10. arjfisiov] corr. ex arjfista m. 1 V. 11. 
HZ] EZ F. 13. o /*»5 — 19. iXux^ctrig] xal ra iirjg PBV 

et F post ras. 1 litt. 16. di] d' p. 17. dnmtsQto p. 

ieti p. sv&siai. tcai p. 19. dsi^ui] seq. i^rig to Q^smQrifia 
V. 22 diax^mai V. 24. itv^s Vp. tio^riv] X eras. B; 
%ol' in ras. m. 1 P. 



184 STOIXEIiiN y'- 

\LEV iottv ij Slcc trou kevtqov, zmv Sh aXKatv Asl 
17 Syytov r^g dta zov x^vt(»ow[ t^s axmTSQOv 
(lE^^cav iet^v, tmv di iCQOs rijv xv^rijv xspt- 
ipiQEiav jrpoefftJTTOvffKii' EV&Etmv iXaxiOt^ fidv 
5 iattv ^ (ittK^v tov ts «TjftEiou xal t^g Siu 
ftdtQov, rmv di alXav asl ^ iyyiov t^g ila 
X^Otrig tijs aaiottQOV idTiv ilatrav, Svo S\ 
fiovov teat KJEo tov eijfttiov jrpoOrefffouvr 
JtQog Tov xvxlov ifp" ixatEQW tijg iXaxlerrjs. 

10 "Eeta xvxXog 6 ABF, xal rou ASF ECX-^ip9m tt 
e^fiEiov ixtog to ^, xal «7c avtov St-qx^etOav ev- 
&£tai ttvES ttl JA, JE, JZ, Ar, ^ffrra S^h ij JA 
Sta Toti xivtQOv. Xiyea, oti tav fiiv repog r^v AEZF 
xoikrfv nEQttpEQEtav 7iQo6mntov6inv Ev9ttmv fiEyieTtj 

16 [liv ietiv 1} Sta TTOU xivtQov ij AA, (lei^eiv 
Sl i] (iii' AE T^s AZ Ti Si JZ rijg ^F, tmv 
Si rcpog tijv &JKH xvQf^v 'JteQifpiQHav wpotf- 
mazovecov Evd^eiiSv IXaxisfq fidv iettv fj ^H 
ftetttlv tov arjfiEtov xal tijs StafiitQov tijg AH, ai 



4 



i 



1. fartv] ^atai B. Port KErTpor add. P: tla%{atrj 3i ij 
fiETK^iJ Tov re <lriji.iiov xal T^e Siafisioou iteoaninTovea i idem 
p, omiaao aqoeTtiTiTOVoa; de!. m. 2; httxieiri (ifv ianv (huc- 
o»(jue 51) ^ (i£TO:|u toC if ariittiov kbI i^e Sia/itTf/ov _" 
scnpto ^ m. 2; aupra tiuv lin. 1 Bcr. a m. 2. d^J d' .. 
^yytiov P. (fRtaTfeuv P, owojiceo] p. 3. iai£v\ PF; oomi 
p; leTtV; iaxai B. 4. llKj;(fltJj — 5. dioftEieou] mg. m,2 
om. p et F, Bupra Eu&ttiiv est ^ m.2. 5. ^ctiv] P" '" 

B. 6. Tm* Si SUmv] om, p, add. m. 2 PF. d a. 

Ifytiov P. 7. BjtciT^eto Pp. ilteTiojv (in ras. m. 1) Jot^» 

p, icTiv] iaTai B. iliJffODii' F. 8. Vaai] P m. 1, P; 

om. p; cvdcibi teat B; r<Taf ev&fiiii Y, F m. 2. tod] tov 

KViov B. 9. nposl '"•rai npiJs p. 10, Po9t ffficg ras. 1 Utt 
V, xbI roO ^Brj om, F. tA^ipi* tf. 12. iivgej P, P 

m, 1, V m. 1; tiwfi TTpoK rof KTJKio» Bp, F m. a, V m. 2. 
In ip»a propositiODO Auguatua bdo arbitoio ordinem Qerbonua 



ELEMENTORUM LIBER m. 



185 



bitns adcidonty maxima est, quae per centrum ducta 
estj ceterarum autem proxima quaeque ei^ quae per 
centrum est^ remotiore maior est^ rectarum autem ad 
conuexam partem ambitus adcidentium minima est, 
quae inter punctum et diametrum posita est^ cetera- 
rum autem proxima quaeque minimae remotiore minor^ 
et duae solae rectae a puncto illo ad circulum aS- 
cident in utraque parte minimae. 

Sit circulus ABFy et extra 

ABF sumatur punctum aliquod^^ 

et ab eo rectae aliquot educantur 

^A, AE, AZ, ^r, et AA per 

centrum ducta sit. dico^ rectarum 

ad cauam partem ambitus AEZV 

adcidentium maximam esse eam^ 

quae per centrum .ducta sit, AA, et 

AE> AZ, JZ > AF, earum au- 

-^ tem, quae ad conuexam partem 

ambitus &AKH adcidant^ minimam esse AH, quae 

inter punctum et diametrum AHf osita, sit, etproximam 




mntanit, sed parmn recte; neque enim Euclides demonstrat 
dA maximam, JH minimam esse omnium rectarum a ^ 
adcidentium, quod tamen inde facile sequitur, quod rectae ad 
SAKH adcidentes omnino minores sunt ceteris. Gampanus 
omisit p. 182 1. 23 : &v (ila — 25. sv^si.mv^ cetera ut nos prae- 
bet Eutocius p. 182, 24—25 et p. 184, 3—4 ut nos legit. 

15. Post dA add. iXax^otri 91 ^ fi,st(x^v tov d arjfisiov xal 
tijs ducfiitQOv t^g AH BFV; idem P {JH pro AH) et p ad- 
dito ts ante J et supra [ista^v soripto ri JH; iXax^otrj dl 17 
fista^v tov erjfisiov xal tijg SiafiitQOv tijg AH ed. Basil. 

16. tijg'] (alt.) t^ FV. 17. 9AKH] K corr. ex Jf V m. 1. 
18. iXaxCatr] — 19. AH] om. PBFVp, ed. Basil.; corr. Gre- 
gorius. 19. dsC] alsl F. 




L 



186 rroiXEifiN y'. 

S% 17 Syyiov T^s ^H iXaxiotijs Hdxrtov iatl t^fi 
ifpoi', 7] [i£v ^K t^g ^A, 17 di ^A r^g ii@. 

JS^A^^&io yccQ ro xsvtqov roti ABF xvxKov xal 
^ero To JW- xal ^jrfgEvxS-raffai' at M£;, itfZ, M-T, MiC^ 
6 iW^, M®. ,\ 

Kal iasl terj iatlv ^ AM zy EM, xoivii «qoO- 
xeis&a 1} MA- ij aga AA (sti istl tatg EM, M/J. 
kAA' al EM, M^ t^g E^ ji«/£ovf's Bisiv xal tj AA 
ttQa T^g EA (isi^av iexiv. aahv, ijttl laii iezXv ij 

10 ME ty MZ, KoivTi Sh 15 MA, aC EM, MA a^a talq 
ZM, MA fffat tiaiv «aJ yavia ^ vno EM^ yat- 
viag tijs vxb ZMA (laiiav ietCv. ^aSig a^a r/ E^ 
pdesag T^g ZA ftetgrav istiv. ofiotmg Si] tfetloftBV, 
ort «ai r[ Z/i r^g TA ftti^av iariv [isyieTti fthr 

15 ttga r] AA, ftf^gov Si ij fiiv AE t^s AZ, 5J Sh -4Z 

Kal iael al MK, K^d t^g M^ (lEi^ovig sieiv, fmj 
Sh 'tj MH t^ MK, loiTfij aga ^ KA i,otJt^s t^s H^ 
fiBi^tav ietiv aezB i) HA z^g K^ iKaxtcav ietiv 
20 ital ind XQtyavov toi5 MAA ial fiiag taiv 'xKevQav 
zffg M^ Svo EV&aCai ivtog SvvaezdQijaav at\ MK, 
KA, at afftt. MK, KA rav MA, AA iXdxtovig aieiV 

1. Si] om. PBFVp, ed. Baail.; corc GregoriuB. fy- 

ytiov P, aed corr. ilmMOiv iazCv PF. anmxifin p. 4, 
ME] corr, ei EM m. 2 V. MF] ME1 <p (non F). T, 

^M P. ioTtV P, lafs] corr. ei la m.l P. B. blU' at\ 
at Si P. r^F] eupra m. 1 P. ilai.v] PBF; e^ai Vp. 

9. Idilv] PF; ioii Tiulgo. 10. EM IB ZM P. tf*] oum 
GreRorio; niioe*Ble&<a PBFVp. ij] om. V. II. tictv] 

PBF; tlal Vp. kbI yojv/a] mutat. in yioWo iJe m. rec. F. 
EM^] E supra m. 1 F. 12. loriv] comp. p; ini unlgo. 
13. ^ir»/ P. 14. JZ P. r^] iJ in ras. V, ici/*] P; 

comp. p; ioTi uulgo. 15. jiiv dE] litt. fi,iv J in raa, p, 
19. maie nai p. JH r^f JK F. J^TT<a*] llttjtiiii) F; 



I 



I 



ELEMENTORUM LIBER m. 187 

quamque minimae z/JEf remotiore minorem; jdK<,dAy 

sumator enim centrum drculi ABF [prop. I], et 
sit M. et ducantur ME, MZ, MF, MK, MA, MS. 
et quoniam AM = EMy communis adiiciatur MA, 
itaque AA = J&M+ MA. uerum 

EM + MA > EA [I, 20]. 
quare etiam >^^ > JEJ^/. rursus quoniam ME^MZ^ 
et communis est MA, erunt EM, MA et ZM, MA 
aequales.*) et L EMA > ZMA. itaque EA > Z^ 
[I,24J. similiter demonstrabimuSy esse etiamZ^^Fz/. 
ergo maxima est A A^ et AE> AZ, AZ> AF. 

et quoniam MK + KA> MA [I, 20], et 

MH^MK, 
erit KA > HA. quare etiam H^ < KA. et quoniam 
in triangulo MA/1 in imo latere MA duae rectae 
MK^ KA intra constitutae sunt, erunt 

MK + KA<MA + AA [I, 21]. 

1) Ne hio quidem emendationes Augusti a mutationibas 
ab eodem in propositione factis pendentes recipiendas esse 
dozi, sed emendatione Gregorii leniore, quamqnam et ipsa ob 
consensnm codicnm incertissima, usus uerba iXaxlerri (liv — 
9^a(kitQ0v T^g A H transposui a p. 184, 16 ad lin. 19 et huic loco 
adcommodauL eodem ducit tenor et propositionis et demon- 
strationifl. sine dubio et transpositio omnium codicum hoc loco 
et interpolatio nonnullorum p. 184,1 (cfr. 4) satis antiquo tem- 
pore a matJiematico imperito ad similitudinem prop. VII factae 
sunt, in quam rursus p. 178, 19 in F ez prop. Vni quaedam 
irrepserunt. 

2) Lin. 10 error codicum iam ante Theonem ex lin. -6 or- 
tus erat. 



iXa<faav 6p. i<ft£ B. Post iatlv add. iXaxictri «9« iativ 

PV; om. BFp, Augustus. 21. awsati^nsaav p. 22. at 

a^a MKj KJ] uqu P. Ante tmv in F lacun. 3 litt. 
iXattovg P, iXdccoves F. 



r 



188 ETOIXEIiiN y'. 

fffT) dh 7j MK Tij MA' Komri kqk tj ^K kon^q t^j 
^A iXdrtcov iarCv. oiioiag dti SeC^oftsv, Stt xal ij 
^A t^s A& iXtttrtov ioriv iXaxiOTTi fiiv apa i\ ^if, 
^Aamav S^ i} (iiv ^K rijg ^A tj 6i ^A rrjg ^&. 
6 Aiyca, oti xal dvo [lovov tdat aao row A aiip,tiov 
npoeTteeovvzat jrpog rbv xvxXov itp' sxarsQtt n^g ^H 
iXaxletijs' awsetdta jrpog tfj MA ev&Bia xal tp 
jrpog Rvtfj eriyiEia t0 M tij vreo KMA yavia Hatj 
ycovia ^ vno ^MB, xal iitst^svx^so i) dB. xai ittsX 

10 iffji ietlv ij MK tji MB, xoivr] Si ^ MA, Svo Sij 
eC KM, M^ Svo tatg BM, MA Heat elelv ixatdga 
ixaziQa' xal ycovia rj VTid KMJ ycovia r^ vjco BM^ 
l'er}' ^dois aga i] AK ^daei tij AB tffjj ietiv. Xiyco 
[ffjj], oti tij AK sv%sia «AAjj fo?; ov n^oeasesitat 

i& n^bs tbv xvxXov dxb roi5 A eijfiBiov. sC yaq Svvarov, 
TtQoemntitco xal sCtco ]j ^dN. ixsl ovv fj ^K xf^ 
^N iotiv i'ari, dXX' ^ JK tfj AB iativ terj, xal 55 
^B apa tfi AN iotiv l!er\, rj Syyiov t^g AH iXa- 
Xtetris tfi dncoTSQOvYieTiv^^i^eri' oitsQ «iJvjfaTov iSsi^- 

20 9ij. ovx apa aXsiovg ^ Svo Heai Jrpog ror ABV 
xvxXov &itb tav A erfiisiov iip' sxdtspa trjg AH ila- 
xietrjs nifoeitBeovvtttt. 

'Ekv aga xvxXov Xr}ip9fi rt erjfistov ixtog, dab di' 
rou arjfisiov Bpog tbv xvxXov Siax&aaiv Bv&etaC ttvBSi 

26 mv fiia (t^v Std toij xsvtqov aC Sl Xomai, cog itvxev. 



I 



L 



1. foT Si] PF; cov iativ tari BV; £v p. MA] MA fm) 
Uxiv p. 2. iXiBaiav F, ut lin. 3. 3. JHJ dH t^b AK 

Fp et V eras. 4. hlaeaiov Bn. IXazzmv S\ v f^»] n Si F. 
5. W] om. Bp. iiTHi] P, F_iii. 1; toai tv»titit V, P m.B; 
iv^^tiat Artti Bp, T. jag Tt^oq F, 9. ■fiavia] om. p. 

10. A;X] EM B, MB p et V e corr. MB] MK Bp et V e 
corr. 11, 6vai BTp. txcti^vB] ixaziQai V. 13. btfi 



ELEMENTORUM LIBER UL 189 

neram MK ^ MA. itaque JK< JA. similiter de- 
monstrabimuSy esse etiam dA< ^&, ergo minima 
est JH, et JK < JA, JA < J9. 

dieo etiam, duas solas aequales a puncto ^ ad 
circulum adcidere in utraque parte minimae ^H, 
construatur ad rectam M^d et punctum eius M an- 
gulo KMJ aequalis L ^MB [I, 23], et ducatur JB, 
et quoniam MK «s MB, et communis est M/ly duae 
rectae KM,MJ duabus BMy MJ aequales sunt altera 
alteri; et L KMJ = BMJ. itaque JK = JB [1,4]. 
dicOy rectae^X aequalem aliam rectam non adcidere 
ad circulum a puncto J. nam, si fieri potest, adcidat 
et sit JN, iam quoniam JK = JN^ et JK «=> J^B^ 
erit etiam JB = JNy propior minimae JH remo- 
tiori; quod fieri non potest [u. supra]. quare plures 
quam duae aequales non adcident ad circulum ABF 
a J puncto in utraque parte minimae JH. 

Ergo si extra circulum punctum aliquod sumitur, 
et ab hoc puncto ad circulum rectae aliquot educun- 

^rius) P, F m. 1, p; tari iczi V, F m. 2; icxiv tarj B. iaziv] 
r, comp. p, iavi uulgo. 14. di}] om. Pp. JK] Kin ras. 
y, BJF', JB(p. 15. TtQog] post xa m. 1 ngog q>; mg. ' yg, 
ytQog xov %v%Xov F. 16. -nLntixa} in ras. V. 17. aUof P. 
JK] KJ F. JB] B e corr. V. 18. aQu] supra comp. 
F m. 2. iyysiov P, sed corr. 19. dna^xiQot p. iaxtv] de- 
leo; cfr.p. 182,9. iaxiv tar}] om. p, August. iSsixd^^ om. 
B, AuKust. Post Iloc uerbum legitur alia demonstratio; u. 
append. 20. rj dvo taui] P et sine dubio F m. 1; ddvvux tp 
seq. ai m. 1 (pro dSvv habuit F ij ^vo), supra scr. /liovov 
hvhBtai. m. 2; r^ 6vo fiovov svQ^siaL taai B, et Y, sed fiovov 
m. 2 supra scr. est; iq Svo cv^&etat nQoanBaovvxat p. nQog 

— 21. ariiisiov] dno xov J arifisiov nQoanBaovvxai. nQog xov 
ABF xvxZov B. 21. xvxXov] m. 2 F. J] corr. ex F V. 
22. nQoaneaovvxut] om. Bp. 23. dno di — p. 190, 9: iXux^' 

axrig] xal xd «J^g PBFV. 26. ixvxs p. 




190 STOIXEIfiN y'. 

■zav [lEv jtQos z^v xoilijv %cQiipiQBiav xpoexatrovSm 
sv&eimv iisyiOtTi fiiv isriv ij J(« tov XBVtpov, zmv di 
aXXtov atl 7] lyyiav r^g dtct tov xevTQOV tijs aitm- 
Ttpoj' (tBl%G)v ietCv, rav S% tQog tijv xvgt^v aEQispi- 
5 Quav jCgoSxmToveinv svQttSv iXaxCatij jiiv istiv ij 
littaS,v Tow tt Gtjnsiov xal r^g SiaftiTpov, tmv 8i 
allaiv asl ij lyyiov T^g iXa%leTri£ t^s ttatorEpdv i6Xiv 
iXattav, Svo 6i fiovov teat ano tov ar^iieiov JtQoe- 
Tiseovvrat, jcqos toi' xvxXov iq>' ixKTEQa tije iXaxietrjg' 
10 oiffp ISei det^at. h 



'Eav xvxXov AijqDd^ji ti Oi^iistov ivtos, axo 
8l Tov ar}^el\ov jrpog rov xvxXov xQoanCitxmOi 
nXsiovg ^ Svo tecci evd^stai, ro Xijip&iv 07][ie1:o9.- 
& x^jiTpov ietl Tov xvxXov. 

"Eota xvxlog 6 jIBV, ivtog Si avtov Orjfisiov 
^, xul dno rov ^ Kpog tov ABF kvxXov TCQoOnmti- 
taOav TtXBCovq rj Svo COai ev&eiai at jJ A, dB, ^T' 
Xiyta, 0T( to ^ ei\\iilov xivtQov iotl tou ^BFxvxXov. 
3 'Eae^ivx^toOav yccQ at AB, BF xal Tetn^^O&iaaav 
SCx'^ *:«t« xa E, Z ffj/ftsfH, xal imlsvx&staai aC E.d, 
Zd SiTix^oieav inX t« H, K, 0, A etjiieta. 

'Easl ovv Cori iativ 17 AEzy EB, xoivi} 8i 13 E^j 

8vo Sij aC AE, E^ Svo rats BE, E^ Coai aCaCv 

i xal fiaeis ij jdA^dosi t^ JB terj' ycovCa «pa fj vnlt., 

a. rmv Si allmv — 10. Seiiai] xaf ti ei^s p. 11 
jtiTtiaiai] TcgoanCmovat Vp. 14. cvStiai taai BV. 

ev»tlai teat BVp. 22. Z J] PBF, V m. 8; i3Z p, \ m. 1. 
K, H, A, e P. 24. Svai Byp. tlaMPFV; tCoC Bp, 

86. K«^ m. a V. 8avis Sqci Y. fffij] P et pOBtaa inBerto 
1«^ F; Cori lctC Vi iera t<nj Bp. 



«4 



ELEMENTORUM UBER m. 



191 



tur, quarum una per centrum, ceterae autem utcun- 
que ductae sunt, earum rectarum^ quae ad cauam 
partem ambitus adcidunt, maxima est, quae per cen- 
trum ducta est^ ceterarum autem proxima quaeque ei^ 
quae per centrum est, remotiore maior est, rectarum 
autem ad conuexam partem ambitus adcidentium mi- 
nima est, quae inter punctum et diametrum posita 
esty ceterarum autem proxima quaeque minimae re- 
motiore minor, et duae solae rectae a puncto illo ad 
circulum adcident in utraque parte minimae; quod 
erat demonstrandum. 



rx. 

Si intra circulum punctum aliquod sumitur^ et ab 
hoc puncto ad circulum plures quam duae rectae aequa- 
les ad circulum adcidunt, sumptum punctum centrum 

est circuli. 

Sit circulus AB T, et intra 
eum punctum ^, et a ^ 
ad j4Br circulum plures 
quam duae rectae aequales 
adcidant JA, JB^ ^F. 
dico, punctum ^ centrum 
esse circuli ABF, 

ducantur enim AB^BF 
et secentur in duas partes 
aequales in punctis Ey Zy et ductae Ej^, Zjd educan- 
tur ad puncta H, K, 0, A, 

iam quoniam AE = EB, et communis est JS^, 
duae rectae AE, EA duabus BE, EA aequales sunt. 
et jJA = AB. itaque L AEA = BEA [I, 8]. itaque 




192 ETODCEliiN y'. H 

AEii yavia tFj vxo BE^ fcij iariv oq&t} aga ixa- 
Ttqa tcov v%o AE^, BE^ yavt.iav' i] HK aQa zijv 
AB T{(t/V8i dixa xal Ttgos dQ9ag. xal iTtai, iav iv xv- 
xf.e} tv&tid tig tv&ttav ziva dixa te xal arpos OQ&ag 
5 TB^tvri, inl zfiq rEftvowffijs iQzl %6 xivzQov tov xvxi.ov, 
iid r^g HK aga iazl zo xivzgov zov xvxlov. dta 
ta avta Sij xal inl zijs ®A iort ro xivzQov tov 
ABr xvxlov. xat ovShv eteqov notvov E%ovaiv al 
HK, &A BV&aZai ^ xo ^ 6i){isi:ov ro /1 aqa 07}iittov 
10 xivzQov iorl tou ABF xvxXov. 

'Eav uQa xvxi.ov f.^g>Q"rj zi erjfieiov ivzog, aab dil 
Tou Ori^Biov ntQos rov xvxKov XQoeninraei jikEiovs ^ f 
Svo taai sv&Eiai, z6 XTjipQ^EV flijfiEfoi/ xbvtqov iezl ] 
rou xvxXov oXEQ SSet Seilai. 



KvxXog KvxXov ov zifivsi xara itlsiova an]- 
(leia ij Svo. 

Ei yaQ Svvmov, xvxloq b ABF xvxlov zov jdEZ'\ 
ZEfiVE't(o xaTa icXeiova atjfieta ij Svo ta B, H, Z, 
20 xaX iai^Evx^eiaai at B&, EH Siia rtfivia&caoav xata 
ra K, A aijfiEia ' xal UTto zmv K, A taig B&, BH 



i 

i 



1. iari V. npo;] PB, F in ras.; yap p in raa., V m. 1; 

ieriv agu Y m. 2. 2. ^] xol jj p, aga] om. p. 3. 

xifivti ii^iK] P; Six" ^£f**t B, Sixa liiivovea V (sei vovaa et 
seq. %ai in ras.), p, F (3();o! tiiijvovai <p). opfl-Kg] o^ag 

tiiivgt Vp et P in raa. Kcti inti] iii raa. F, seq. in mg. 

tranBeuDt. mq:1 ixti — 5. tejxv^] mg, m. lec, F. xe] in 

flne im., in mg, add. ftvjj m, 2 B. 6. lifiv^l tifivti, F v. 
i^k] om. F? iaiiv F. 6. iativ B, 7. ioiiv P. 8, 
ABF] om. p. «okIou] tn. 2 Fj om. B. 12. Ttfoaniiiraai 
— 14 KiJMlou] Koi TB f^^s p. 12. -TO^nroKrt in raa. F. 

13. (^ffjitii ftwi B, 14. Seq, alia demonstratio, de qna n. 

appendix, . 16. la' F, eed a etras. 18. JEZ] corr. ex 



ELEMENTORUM LIBER m. 



193 



aterque angulns AEJy BEJ rectus est [I^ def. 10]. 
ergo HK rectam AB et in duas partes aequales et 
ad angulos rectos secat. et quoniam, si in circulo 
recta aliqna aliam rectam et in duas partes aequales 
et ad angulos rectos secat, in secanti erit centrum 
circuli [prop. I corolL], centrum circuli in HK erit. 
eadem de causa etiam ia SA erit centrum circuli 
ABF. nec ullum aliud commune punctum habent 
HKy SA rectae ac A punctimi. itaque A centrum 
est circuli ABF. 

Ergo si intra circulum punctum aliquod sumitur, 
et ab hoc puncto plures quam duae rectae aequales 
ad circulum adcidunt^ sumptum punctum centrum est 
circuli; quod erat demonstrandum. 



X. 

Circulus circulum non secat in pluribus punctis 

quam duobus. 

nam, si fieri potest, 

circulus ABF circu- 

lum AEZ m pluribus 

IB secet punctis quam 

duobus By H, Z, ®, 

et ductae BS, BH 

in punctis K, A in 

duas partes aequales 

secentur, et b, K^A ad 

BSy BH perpendicu- 

JEH m. 2 V. 19. Z, 0] corr. ex ®, Z m. 2 V. 20. B9, 
BH] P; B9, HB F m. 1; BHy OB F m. 2; BH, BO BVp. 
TSTariaJ&maav 8i%a p. xsxitria&toaav P. 21. fi6^, BH] 

BF, V m. 2; BH, B9 Pp, V m. 1. 

Baolidet, edd. Heiberg et Menge. 13 




194 STOIXEISN y 



ngbg OQ&ag a%9Bt6ai aC KF, AM Si^^x&taaav inl 
A, E eyiieia. 

'ExeI ovv iv xvxXm rra ABT iv&ila ttg ij AP 
Bv^tlav riva tijv B& Si%a xal ;tpos OQ&ag ziitvsi, 
5 iicl TJjs AF «p« ietl z6 xivxQov tot) ABV kvxXov. 
xdXiv, inel iv xvxlp ra avta tp ABF avd^EVa zig 
ij NS BV&Btav tiva tijv BH Sixa xal jrpog dp&ag 
tiftvBi, inl tijs NS aqa ietl t6 xtVrpov rov ABF 
xvxXov. iSBix&ri Si xal ixl t^g AF, xal xat' ovShv 

10 Svfi^aXXov6i.v aC AF, NS BV&Biai i} xazcc tb O' tb 
O &(fa a^fiBtov xivtQov iazl tov ABFxvxXov. o^oims 
dij dBi^ofiBV, 0T( xal tou AEZ xvxXov xbvzqov ietl 
TO O ' 6vo d^a xvxkav teyuv6vta>v aXX-^qXovg tav 
ABF, AEZ tb avTO iOTi xbvtqov ro O" okbq ietlv 

15 aSvvatov. 

Ovx aqa xuxAog xvxXov tifivci. xata aXeiova Oij- 
fiBta ij dvo' oxBQ iSBi dBtiat. 



} 



'Eav Svo xtlxAoi i^dntavtat aAAijArav iv- 
20 Tos, xaX Itjfp&ji avtmv ta xivtga, tj ixl r& 
xivtpa uvtmv ijttttvyvvfiBvr] sv&tta xal ix- 
/JaAAopfVii inl t^v avvtt(pi}v nBeBtta 
MXeav. 

Avo yccQ KvxXoi oC ABF, A^E iq^amie&aeav 
2a dXX^^Xmv ivxbs xata to A etjfielov, xal i(Xri<p&(0 



\ 



1. KT, AM] litt. r, J in raa. m. S P; KA, FM V, 
corr. m. 1. 2. A, E] ia raa. p! AE, B.A P. B. tp] 

corr. V ro. a. 4. Slia le BVp. iiu;]^sni>ra m. 2 F. 
7. St%a lifivri xo) jtgiis 6g9ae p. Aate ie&ds raa. 1 Utt,T. 
8. To «iVreo. leti BVp- 9. yai] (prius) m. 2 V. 10. 

tv&itai] ow. p. 17] P, F m. 1; dU^lais ^ BVp, F m. 2. 



1 



^M ELEMENTORUM UBER 111. 195 

Iftres ducantur KF, A M et educantur ad A, E puucta, 
iam quoniam in circulo ABF recta aliqua AF aliam 
rectam B& in duaa partes aequales et ad anguloa 

txeetos secat, in AF erit centrum circuli ABF [prop. 
1 coroll.]. ruraus quoniam in circulo eodem ABF 
ncta quaedam NS aliam rectam 5 ii in duas partes 
aeqiiales et ad angulos rectos secat, in N S erit cen- 
trum circuli ABF [id,]. sed demonstratum eat, idem 
in Ar esse', nec usquam concurrunt rectae AF, NS 
ezcepto puncto O, Oigitor centrum est eirculi ABP. 
simiiiter demonstrabimus , O etiam circuli z/£Z cen- 
trum esse. itaque duo circuli inter se secantes ABF, 
^EZ idem habent centrum O; quod fieri non potest 
[prop. V]. 

Ergo circuluB circulum non secat iu pluribus pune- 
I qnam dnobus; quod erat demoustrandum. 

XI. 
Si dno circuli intra continguut inter se, et sumptn, 
mt centra eorum, recta centra eorum coniungens 
iducta etiam*) in punctum contactus circulorum 
let. 

□am dno circuli ABF, A/JE intra coutingant 
ber se in A puncto, et sumatur circuli ABF cen- 



1) Mmoe recte in B post l*^itiXoaivri iaterpungitnr; 
a Eaclidis potiuB ix^allo^ivi] »al postulat; 



.4. 10 O] om. P. 14. ititivl om. p. 1 

Sequitnr alia demonatratio, n. appeiidix. 

9. tvtiis] mg. m. 1 P, 2Q. *ai li]q;9^ nt 

«w?«rj om. B. 21. «b/] om. V. 22. maUxat'] 

■ »8, m. 3 V, 34. anxiaftmeav Theoii (BF Vp). 

13* 




196 STOIXEiaN y'. ■ 

(liv ABF xvxkov x^vTQOv z6 Z, lov S^ jitJEzo IP 
i.eym, oti ij dnb toO H iitl t6 Z ini^Evyvv(tdvii sv&tUt 
ix^ttllofisvt] inl tb A TiEaetxtti. 

Mi] yap, aXX' ti dvvttiov, MXthoi ats V ZH&, 
5 Xttl iitB^evx^'^'^^''' "^ -^^j -^^- 

'Extl ovv at AH, HZ t^j ZA, rouTEUri rrjg Z9, 
(iti^ovis eIoiv, xoiv^ d^tip^^a&a rj ZH' Koiavi apa ^ 
AH lom^s ■CTJff H® fLstifav ieziv. tst} di tj AH zy 
HA ■ xal 7] HA aga ttjs H& [isLlmv iozh i} iXaTTeiv 
10 r^s [ie(^ovos' oxEff iexlv advvttTov ovx uQa 17 aTch 
Toii Z inl Tc H ijti^Evyvv(iivij ev&eta ixrbg «ESslTaf 
xata To A aQa ixl r^g flvvttipiis JtaeEiTai. 

Eav aptt Svo xvxXoi itpaatiavTat dXl^^Xav ivroSf 

[xai lt}ip9)j avTKiv tu xivzpa], ij ial ra xivTQa avxmv 

15 iat^Evyvvftivi] ev&ata [xul ix^aXlofiivti] iid r^ Ouva- 

cpijv nsaEttai rmv xvxXov' 0]ig(i iSei SBt^ai. 



'Eav dvo xvxXoi i^aXTavtai aXX-^lav i%~ 

ros, ^ inl ta xivrga avtmv iai^evyvvfiivt] dia 

£0 T^^ imaip^s iXevOBTai. 

AvQ ya(} xvxXoi, aC ABV, A^JE icpajitie^aOav 
alX^Xav ixtbs xara ro A etjnstov, xal siX^^ipQio rov 
(liv ASr xivTffov Tt) Z, Tov 3e A^E rb H' Xdyaf 

1. fiBv] om. B. 10 nivcsov t6 P. 3. A ariiuiov FT, 

P ». rec. 4. ZH0] ZB F, H supra scr. m. 2. 6. at] ij 
P. ZA] in rae. m. IV. t^s ZA] mg. m. 1 P. TOviiouv 
P. 7. ilmv] P; elai nvlga. ZH] H ia ras. V. 8. &ij 

ai _ 9. latit,] mg. m. 2 B iloTt). ("ffij dh ij AH ri H^] in 



:(H mg. m. 2 , . 

IHi PB, F m. 1. V ra. 1; JH p, : 

1, F m. 1, V - ■ " " 



■. H^] PB, F m. 1, V in. 1; AH p, F m. 3, V m. 2. Hdo- 

m- Fp- 10. iaUv] PF; om. BVp. j{\ supra m. 1 P. 

1. Port Ixrds add. t^s *azu 10 A ewaip^i Tbeou (BFVp), 



ELEMENTORUM LIBER Ul. 



197 



tmm Z, circnli aiitem A^dE centriim H [prop. I]. 
dico, rectam H, Z coniungentem prodactam in A ca- 
saram eaae. 

ne cadat enimj sed si fieri potest, cadat ut ZH& 
i dncantur AZ, AH. iam quoniam 

AH-{-HZ>ZA p, 20], 

h. e. AH-\- HZ > Z@, subtrahatur, 
quae communis est, ZH. itaqne 
AH>H&. sedAH=HJ. itaque 
etiam H^ > H&, minor maiore; 
qnod fieri non potest. itaque recta 
Z, H coniungens extra non cadet. 
e in A inpunctum contactus cadet. 
duo circuli intra contingunt inter ae, et 
Bnmpta erunt centra eorum, reeta centra eorum con- 
iungens producta etiam in punctum eontactus circulo- 
rum cadet; quod erat demonatrandnm. 

XII. 

Si duo circuli extriusecus contingunt inter se, recta 

i coniungens per punctum contactus ibit. 

nam duo circuli ABF, AAE extrinsecus eontin- 

flit inter se in puncto A, et sumatur circuli ABF 

latnaa Z, circuli autem A^AE centrum H [prop. I]. 




IS. KOTa To A &^a ijil x^g evvciiffii nrEaEiiai] P; 
i' aixiti a^B p; i«' uvi^; B, a^a add. m. 2; ln' avTT\v aga 
■ l-' avxois aga F. 13. iifditxiBvxai] amaivTai. PB, et F, 
' aupra m. 1. 11. nal ItjifiQij avi^v la xEvi(>a] ing. 

;; om. PVp. 16. xoJ UpaUoiiivT]] om. PFp, 16. 

x£v HCiUwv] om. p. Seq. alia demonatxatio; u. appendii, 
17. ifl'1 om. ^. 18, aTnaivtui Theon (BFVp). 19, ev9fia 
eia BV, F m. i. 28, ABF] e corr. F. Dein ^v-xXov add. 

PV, V m. 2, 



t 



198 



ETOIXEiaN y 



OTi 7} KJto xov Z iTtl ti> H ini^Bvyvv(idvti Ev&ata (JtA 
TJ5s xaxK zo A iita<prjq ikEv6£rai. 

Mij yap, kAA' e^ ^vvaTov, iQy^ie&ta rag ^ Zr^H^ 
kk\ ins^tvxQiiiSKv ul AZ, AH. 
6 'BjTil ovv to Z Oijftfrov xivTQOv isrl xoO viRJ 
xvkXov, i'6rj iazlv ij ZA r^ ZF. Jtakiv, inel 
a^liBtov KevtQov ietl rot! A^E xvxlov, [6rj iaxlv ij 
HA t^ HJ. ideix^ii Sl xal rj ZA ry ZT fffi)- at 
aQa ZA, AH rats ZT, H^ i6ai daiv &6tb oii? ij 

10 ZH tmv ZA, AH [iBi^av iatCv aXXa xal iXttTTotv 
oycE0 iailv ddvvatov. ovx apa ij dao rou Z ixl to 
H ETit^Bvyvvfiiv'^ EV&Eta 6ia tijs xara to A ijtttipijs 
ovx ikBvOetai ' Si avr^s aga. 

'Eav apa Svo xvxloi ifpoMtavrat akXriKmv ixroff, 

15 71 bJii ta xivTQa avtav iict^evywiiBvrj l8v9£ttt] Am^i 
c^S inaipijg iXEv6ETaf oaep SSsi det^at. 



1 



)vx itpaxtstai xattt xXsCovtt- 
idv TB ivToq idv te ixtoQ 



KvxXog xvxXov 
67ntEta ij xa&' ev 
itpdxTTiTai, 

Ei yuQ rfuvaioV, xvxlos o ABF^t xvxXov tov 
EBZA ifpaatB'69a> Jtporepov ivTOS xard JtkBiova 6tj- 
fisia jj ^v ra ^, B. 



2. ■^xa 10 41 BUiira m. 2 Y. 4. -iZ] lA P. 6. Zj(] 
^ V. 8. AH F, Ante Hd 1 litt, eras. F. 9. ZT] Z 

V, corr. es r to, 1. HJ] JH Pp. 10. ^lairto*] il«(i<T«M' 
F; 11 «««01* V. 11. iis^iv\ om. p. 10^] rd B. 18. 

H] M qi (non F}. 13. aixiiV f. op«] om. B. 14. 

'Eb»] av V. 16. ij ^B^ in raa. m. 8 V. fv»tXa 9(1»] 

PBFV. U. ^ti* Bpa — 16. llfiofzaC^ om. p. 16. 

Sitff IJst a*r£Qi] :— UF, 17. ly'] ic' F; corr. m. 2, 



I 



\ 



ELEBfENTORUM LIBER m. 



199 



dico, rectam Z^ H coniungentem per punctum con- 
tactus A ire. 

ne eat enim^ sed si fieri potest^ cadat ut ZFdH^ 
et ducantur AZ, AH iam quoniam Z punctum cen- 
trum est circuli ABF, erit Z-^ = ZF. rursus quo- 
niam H punctum centrum est circuli AjJE, erit 

AH:=^HA. 
sed demonstratum est^ etiam 
Z^ = Zr. itaque 

ZA + AH^^Zr+HJ. 
quare ZH > ZA + AH. uerum 
etiam ZH<ZA + AH [I, 20]; 
quod fieri non potest itaque recta 
Z, H coniungens extra punctum 
contactus A non ibit. quare per 
A ibit. 
Ergo si duo circuli extrinsecus contingunt inter se 
recta centra eorum coniungens per punctum contactus 
ibit; quod erat demonstrandum. 




xm. 

Circulus circulum non contingit in pluribus punc- 
tis quam in uno; siue intra siue extrinsecus contingit. 

nam si fieri potest, circulus ABF^ circulum 
EBZ^ prius intra contingat in pluribus punctis quam 



18. ovx] supra m. 2PV. xara xd V, sed corr. 19. hz6i\ 
ivxos itpanxrixai P; Ixrog B et V m. 2 {ivxoq m. 1). ixrdgj 
hxoq BV. 20. iq)anxrixai\ om. P. 21. ABTJ] ABT lac. 
1 Htt. 9>. 22. £Z, Z^ P, corr. m. rec. anxiad-m Bp et 
F m. 1 (corr. m. 2). 23. Jy B] B, ^ Pp. 



200 XTOIXEIflS v'. 

Kal elXrtp&a zov (ilv ABFji xvxKov xivxQov 
H, rou da EBZ^ ro &. 

'H Kpa aao tov H inl to & ini^tvyvviidvij iscl 
B, A nEeeiiat. niictiza ms V BH®^. kkI iTtel to 
6 H eijittiov xivzQov iszl row ABV^ xvxlov, lai] ie-ulv 
{) BH zrj H^' (itit^v Spa ^ BH rijg @jd' xoll0 
«p« /igi^iav rj B@ rijg @z/. nakiv, insl ro & erjfielov 
xivTQOv icxl Tov EBZid xvxXov, tarj ietlv ij S& rj 
@J- iSU%&ri Sl avzijg xal aoXlp itii^mv ontg dSv- 
10 varoV ovx «pK xvxiog xvxkov ifpajcztTai ivtog xatt 
nXfiova er^fteia ^ ev. 

Aiya tfi}. ort ovSl ixtog. 

El ya^ Svvatov, «vx^os o AFK xvxkov xov ABTA 

itpantis&a ixxhg xaza nXttova erifitta ij Sv za A, Pj. 

15 xal intlivx&io 7] JF. 

'Entl ovv xvxXiov tav ABVJ, AFK tiXrjnzai ii 
T^S nsQKptQtiag ixaziqov Svo tvxovra Grifitta xa 
r, {j inl za 0i^fi£ta int^tvyvvfiivr] tv&sta ivzog Sxa- 
ri^ov ntattzai ■ akXa xov fiiv ABF^ ivtbg insatVf 
20 xov Sl AFK ixzog' osfp azonov ovx aqa xvxkog 
xvxKov itpdnzexai. ixzog xata nXiiova Si]fitta ^ ftf, 
iStix^^J Si, ott ovSh ivzog. 

KvxXog aga xvxXov ovx iipanztzai xazd jiXtii 



1 



1. ABF^I P, F in rafi., V 
B, V m, 1, p m. 1. 3. &] i 

tv6tla iitC Vp, F m, 2. 4. mTMiffl 9. 6. BH'\ (alUl 

JH P, corr. m. rec. i^e] oorr. as tj m. 2 P. BJ\ j"™ 
108. 1 litt., i^ poBtea insert. m. 1 V. 8. laxiv lait V. 

oxtQ iariv F. 12. tf^] m. 2 V, 13. Svvaxov yag p. 

Arx} AKr Fp, ATKA B, P m. 2. A&iir Bp; .JT litt 

in raa. V, eras. F. ^FJI] AKF p, AFKA B, Pm.3, T ia 
raa. m. 2. 17. *iJo] supra aer. m. 1 F. id vi — 18: ffij- 

(i£io] mg. m. 1 P. 18. ij «ea P. tu avxi B. 19. ASiHr 




ELEMENTORUM LIBER IIL 201 

uno ^y B. et sumatur circuli ABF^ centrum Hy 
circuli autem EBZJ centrum 0. 

itaque recta H, coniungens 
^/ \ in J5, ^ cadet [prop. XI]. cadat 

ut BHSjd. et quoniam H 
punctum centrum est circuli 
ABFJ, erit BH^H^. ita- 
que BH>SJ. quare multo 
magis BS> 9J. 
rursus quoniam S punctum centrum est circuli 
EBZdy erit BS^Sjd. sed demonstratum est, ean- 
dem multo maiorem esse; quod fieri non potesi ita- 
que circulus circulum intra non contingit in pluribus 
punctis quam uno. 

dico igitur, ne extrinsecus quidem hoc fieri. nam 
si fieri potest, circulus AFK circulum ABFd extrin- 
seeus contingat in pluribus punctis quam uno A, F^ 
et ducatur AF. iam quoniam in ambitu utriusque 
circuli ABFAy AFK duo quaelibet puncta sumpta 
sunt Ay Fj recta ea coniungens intra utrumque cadet 
[prop. 11]. sed intra circulum ABFA et extra cir- 
culum AFK cecidit [def. 3]; quod absurdum est. ita- 
que drculus circulum extrinsecus non contingit in 
pluribus punctis quam uno. demonstratum autem, ne 
intra quidem hoc fieri. 

Ergo circulus circulum non contingit in pluribus 



Fp. inBGi Vp. 20. AFK] K in ras. m. 1 P. 21. Iq?a- 
'^erai B, V snpra scr. m. 2. 23. ovx] snpra scr. F. itp' 
diffeteu BF, V e corr. m. 2. 



202 STOIXEIHN y', 

T«( ■ 07CBQ b8bI dBt^ai. 



ixzog i 



'Ev xvxla ctC taai sv&Etai- iflov aj[^j;otfffiflj 
6 ttjco Tov xivTpov, xal at teov axix^'^' 
Tov xivxQov leai «AA^Aaig tialv. 

"Eexta xuxAog 6 ABT^, xal iv avrip [0ai, Bv&tH 
lOTfoaav aC AB, FJ- Xiyw, ort al AB, FjJ [ao% 
aaixovaiv ^wo rou xivx^ov. 
10 Eikrjfp&m yiiQ to xbvtqov xov ABFJ xvxkfA 
xal Eaxm xb E, xal aito xov E inl zas AB, F^i t 
#£T0( ^%%aGav at EZ, EH, xai ins^Evx^''^^'^ ' 

AE, sr. 

'Exel ovv Bv9Bta xtg Sia xov xivxqov ii EZ i 
15 Qeldv xiva ;*^ Sia xov xivxQOV xijv AB npos 6p&&t 
xifivet, xal SCxa «urijv xBfivai. i'6t] aga ^ AZ ty ZB^ 
SinX^ apa rj AB t^g AZ. dia xcc avxa Sij xal ^ f^ 
T^s FH iezt Si^li)' xaC iexiv lei] ii AB xf} V^' 
Ceji aQa xal ij AZ xtj VH. xal ixBl CeT} iaxlv i] AE 
50 xjj EFi teov xal ro ajio r^g AE za dno x^g EF. 
dlla rp fiiv dxb xijg AE Cea xd d%b xav AZ, EZ' 
OQ&i] ya(f ij jrpog xp Z ymvia' ra Si dico t^s EF 
Cea xa dab ttov EH^ HF' 6&&ij yaQ ij sipos xa H 
ymvia' xd aga dab xmv AZ, ZE Cea iexl toig ttitb 



1. %a»'] om. PBPVp. l»TOs] tHioe BV, i%t6s] Ivtis 
BV. PoBt ivTos in F est j. 2. 01119 ^^" Sfitai] ir^ BF, 
om. P. 3. t«'] (s' F; ootr. m. 2. 4. Iv] inter s et »- 1 Utt. 
eraa, P. 7. ABJT p, 8, 0« at AB, V^] P; 0« Theon 

(BPVp), 10..^B^rp. la.afEZ — ^)tE£tc;[&aioB*]nig, in,lP, 
13, AE] Utt. A m roa. m. 2 V. ET] FE Pp. 16, xinvH] 
(alt) «(»t» FV, ZB] BZ P, Z» 9 {uon F), 18. latt\ 



ELEBfENTORUM LIBER m. 303 

panctis quam in uno, siue intra siue eztrinsecus con- 
tingit; quod erat demonstrandum. 

XIV. 

In circulo aequales rectae aequali spatio a centro 
distant, et aequali spatio distantes a centro inter se 
aequales sunt. 

Sit circulus ABF/I, et in eo aequales rectae sint 

ABjF^. dico, ABfFjJ aequali spatio a centro distare. 

sumatur enim centrum circuli ABF^ [prop. I], 

et sit Ef et ab E 9A AB^ FA perpendiculares ducan- 

j tur EZ, EH^ et ducantur AE^ EF. 
iam quoniam recta quaedam per 
centrum ducta EZ aliam rectam non 
per centrum ductam ^^ ad angulos 
rectos secat, etiam in duas partes 
aequaleseamsecat[prop.IU]. itaque 
AZ = ZB. ergo AB = 2 AZ, 
eadem de causa erit etiam FA = 2 FH, et 

AB = FA. 
itaque etiam AZ = FH.^) et quoniam AE = Er, 
erit AE:^ = En. uerum AZ^ + EZ^ = AE* (nam 
angulus ad Z positus rectus est) [I; 47], et 

EH^ + Hr^ = Er^ 
(nam angulus ad H positus rectus est) [id.]. quare 

1) I %oiv. ivv, 6, quae cum genuiiia nou sit, Euolides 
U8U8 erat I %oiv. ivv. 3. 




icTiv B. 19. insn inl <p (non F). 20. J£] mutat. in VE 
Vjm. 2, FE in ras. B; eras. F, in quo seq. ymvov (post lacun.) 
xi^ymvtp. £r] il£ B et e corr. V; in F euan. 21. (tictr] 
om. B. tcu imi B. £Z] Z£ Pp. 23. taa iatl B. 
HF] corr. ex FH V. H] Z tp (non F). 24. iaxiv P. 




i 



204 STOIXEIiiN y'. 

TiDV FH, HE, mv ro ano rijs AZ laov itftl i^ 
t^ff FH' i'at] yup ietiv ^ AZ t^ FH- kotxov aQu 
ro ano Tije ZE za a%b r^ff EH l6ov iiSziv ' fffij a(fa 
ij £Z tfl Bii. iv di xvxXa i'aov aadxtiv K3tb row 
5 x^vrpou su&frat liyovrai, ozav aC anb zov xivTQOV 
in avTas xa&troi ay6(iEVtti fffori meiv al kqk AR^ 
ViJ teov aTci%oveiv areo xov xivr^ov. 

'AXXa tfiy at AB, FA evd-etat teov amiiriaaav a: 
tov xivT^ov, Tovrieriv teij ^atta 17 EZ tij EH. Jiiyi 

10 oTi far} iatl xal ij AB tfj FJ. 

Tav ya^ avzmv xaxaexEvae^ivTav bnoitog 81 
iofitv, oT( dtjrA^ iativ ^ [i£v AB r^g JZ, ^ di VA 
r^S rjf ■ xa\ i-HBl fOi? iatlv i} AE r^ rE, teov iotl 
To «310 tijs AE ra ajtb rijs FE' akJ.K rp ^'kv dab 

16 tijq AE tea ietl ta dxb t6v EZ, ZA, ta Sh dxb 
tije rE tea ta dxb tmv EH, HF. t« Kpa dxb tmv 
EZ, ZA tea ierl rorg dab tmv EH, HF- av ro dnb 
T^q EZ t^ dnb T^s EH iativ ieov ' fffjj ya^ rj EZ 
r^ EH' i,ombv «pa ro dnb tijg AZ teav iarl ta 

20 dnb i^s FH' tarj «pn ^ AZ t^ FH' xai ieti tijg 
fiiv AZ Sml^ ^ AB, tijs Sl FH tfmA^ tj FJ- raij 
Kpa rj AB v^ Tz/. 

'Ev xvxXa ttpa «t teai, Ev&£iai teov dxi^oveiv dic6 
rov xivzQov, xaX at teov dnE%ovaai. dnb tou xivtgov 

25 teai dXli^laig Eteiv ' oxbq Idn dEt^ai. 

S. rm] P, V m. I; Joijim rcS BFp, V m. 2. Ante t^ in 
V est roo» hti. &o« iaiiv] otn, V, ^011»" Caov Pp. SQat 

xol 15 P. 4. EZ] ZE P. 5. af] om. p. B. alla 3^] 

jtdXiv Bp. 9. EZ] con: ex JZ m. 2 P. 10. toiCv P. 

11. ofwAie ari BFp. 13. ioiC] om. BV, ^ai p, lor/i' P. 

14. aXltt\ m.2 V. 16. torC» P. 17. ftfa] faoi ip. lacfr 
P, ro airo tijq] mg. m. 2 V. 18. £2] P, F m. 1 i EH 

Bp, F m. 3, V mg. m. 8. Deinde iu p aeq. teov Ini. *f] 



i 



ELEBfENTOBUM LIBER m. 206 

AZ^ + Z£? = TH^ + HB?. 
sed AZ^ «= rfl*; narn AZ = Tif. itaque 

Z£« = JEHl 
quare £Z = EH, in circulo autem aequali spatio a 
centro distare dicuntur rectae, si rectae a centro 
ad eas perpendiculares ductae aequales sunt [def. 4]. 
ergo AB^ T/l aequali spatio distant a centro. 

Uerum rectae AB^ T/l aequali spatio distent a 
centro; h. e. sit JSZ = EH. dico, esse AB = TJ. 

nam iisdem comparatis similiter demonstrabimus 
esse AB = 2 AZ, TJ = 2 TH. et quoniam 

AE = TE, 
erit etiam AE? — TE?. uerum 

EZ^ + ZA^ = AB? P, 47], 
et EH^ + HT^ = TE? [id.]. itaque 

£Z« + ZA^ = JSfl» + Hr*. 
sed JSZ* = £H*5 nam EZ = £tt itaque 

^Z» = TH\ 
quare ^Z = Fif. et erat 

AB = 2 AZ, TA = 2 TH. 
ergo AB = T-^.^) 

Ergo in circulo aequales rectae aequali spatio a 
centro distant, et aequali spatio distantes a centro 
inter se aequales sunt; quod erat demonstrandum. 

1) I Hoiv. ivv. 5. Enclides ad I xoiy. kvv. 2 prouocare 
poterat. 

' < ■ ■ 

corr. ex x6 m. 2 V. Efl] P, F m. 1; EZ BVp, F m. 2. 
icxw tcovl PBF; om. p; tcov ictiY. Deinde seq. inV: tw 
dno T^s EH pnnctis deletum (itaque V a m. prima habuit 
idem quod P). EZ] ZE p. 19. saTtv P. 20. affo] 

corr. ex yuQ m. 2 V. iaxiv P. 21. 17] (prius) supra m. 1 
V. rj] AJ q> (non F). 23. afj om. P. 25. dX^Xotg P. 




'Ev xvxXw HByiez'^ fiiv 
Sl aMfnv del ^ iyyiov T0J3 
t£Qov ftftjrar iexlv. 
& "Eera xvxXog 6 ABF^, SidiisTQog Si kutow iet0. 
71 A^, XEvrgov Se to E, xal lyyiov fiiv tijg A^^ 
Sia^itQov fffrra ^ Sr", dnmxEQOv S\ i) ZH' Acye>, ow 
fityietr] iUv ieriv rj A^, p.Eii(ov Si ^ BF tfjg ZH. 
"Hx&ioeav yap «reo row ExivtQov inl tag SF, Z3 

10 xd96toi at E®, EK. xal iml Syyiov (liv rov xivTtfov 
ietlv 7} BV, dnmtHQov S^ ^ ZH, ^eitfov aQtt 17 EK 
T^g E&. xtie&m rii E& tOij tj EA, xal Std rou .^H 
ty EK npog OQ&dg d)^&Et0a ij AM dtij^&oj ixl to N^ 
xal inE^svx^amv al ME, EN, ZE, EH. "1 

16 Kal ixEl tet} ietlv t/ E® Tr} BA, ieij ierl xal ^i 
BT tfj MN. jidXiv, insl fffij ietlv ^ jth' AE t^ EM, 
fj Se EA t^ EN, ^ aga AA tatg ME, EN tet) iettv. 
dXl' ai filv ME, EN T^e MN (istiovig eiffiv [xal ij 
AA ti^s MN fititiov ietiv], tef} 31 i) MN ty BF- 

20 ij AA ttpa r^s BP (lEi^mv iSTiv. xal iitel Svo at 
ME, EN Svo tttlg ZE, EH teai eieiv, xal ytavia 
r} imo MEN yaviag T^g vno ZEH fiEi^mv [ieTiv], 
§deig dffa ^ MN fideEag tijg ZH (lEi^oiv iotiv. dXXa 

1. (£' erae. F. 2. iiiv iativ BTp. ^ 3. SiJ 8' Bp. 

tyyttov P, aed coit,, nt lin, 6. 10, i^s dia zov V. axai- 
itQO) p. 5. ^aim] oni. p. T, Post SiatthfOv rae, 3 titt. F. 
9, E] Bupra m. B V, 12, E0. Heiaffto rji E©] mg, m. 2 

V. Ha! xe^a&o) B, faij i; EA] in raa. ante lacnnam llitt. 
V. U. EM BYp. EZ p. HE P. 15, laxf\ ImCv 
PBP. Ifi. (aM m. 2 V. 17. E^] J m, 2 V. EIV] " 

(alt.) N e corr. V m. 2, 18, alla P, niv) om. BTp. 

EiV, EJtf F; EM, EN p. fitiSous P- e^o.»] PBF; tlat 

Vp, 19, upu i^c p. loi^ T. iot] 6i rj — 20: Fictja» 



^^ 



ELEMENTOBUU USER UI. 307 

XV. 
In circnlo mazinia eat diametnis, ceterarum autem 
prozima qaaeque centro remotiore major est. 

Sit eircnlue ABF^, et diametmB eiiiB Bit Aj^, 
centrom antem E, et diametro A^ propior eit Br, 
remotior autem ZH. dico, mazimam esse AA, et 

jjr> zu. 

ducantur enim a centro E ad BF, ZH perpen- 
dicnlares ES, EK. et quoniam BF centro propior 
eet, remotior autem ZH, erit EK> EB [def. 4]. po- 
natnr EA » ES, et per A ad EK perpendicularis 
ducta AM educatur ad N, et ducantor ME, EN, 
ZE, EH. et quoniam E0 — EA, erit 
etiam Br= MN [prop. XIV]. nu-Bus 
quoniam AE == EM et E^ = EN, erit 
\aA = ME + EN. sed 

ME -f JEN > MN [1, 20], 
' et MN=Br. itaque^) AA>Br. et 
quoniam duae rectae ME, EN duabns 
ZE, EH aequales sunt, et 
LMEN>ZEH, 
erit MN> ZH [I, 24]. sed demonstrandum est 

1) Cum aoa lin. 19 in detemmo boIo oodioo Bematum Bit, 
coaiectarae deberi nidetar-, qDore pnto, uerba mtl ^ A^ c^c 
MN litiitov iaxlv gloBsema antiquum esse. idem de nerbis 
nal 7t Sr tqc ZH ^ttitmv Imiv p. 208,1-2 indico. 



i<rtiv2 om. BVp. ao. i^e] ifli F. 21. JWE] BM p. 

tteiv'] PF; ilai uulgo. 22. htlv] om. P; oomp. Fp; ivti 

BV. 23. oli' F. 




r208 rroiXEiiiN y'. ^f 

ri MN ty BT M«x*)] fffj; [xal rj BV ZTJg ZH fU^^ 
Sh 71 Br TJjg ZH. ^ 

'Ev xvxl^ ftpa fisyiazTj fiiv iativ fj Siafietffog, 
5 Tav 6i aiXav «f! rj Syyiov roi^ xivtgov rijs dmoteffov 
liei^cav iativ ozsq ISei Set^at. 



'H tfj StafiitQp tov xvxlov repos vQ^ag aX 
axQag uyofidvT] txtog Jicastrat loi) xvxXov, xaln 

10 eig tbv (lEtaiv tontov r^g tt ev^sias xal rqtt 
nsQiipe(/eiag itEQa ev&tia ov aaQ£(iJcsaEttttt.t 
xai ^ (ihv tov ijfiixvHiiov yavia anaa^ 
viag oisiag Ev&vyQafi.^ov fteigwv iativ, i} ilM 
Aomff ikiittcav. 

15 "Eata XTjxios 6 ABr nE^l xivxQov zo /i i 
Siaftet^ov t7]v AB' Kiyto, OTi if aao TQV A t^ A^ 
nqog 6i)^as an axQag ayofLivr} ixtog nsasttai Toi; 
xvxAov. 

Mij yap, aAA' ei Smratov, atittiToi ivtbg rog ij rA, 

20 xal iTtelEvx&co ij JF. J 

'EksI tay} iatlv ij ^A t^ AT, taij iatl xal yaivtf^ 

i] vnb AAF ycovia tfi vnb AF^ii. oq^ 6h ij vxit 

AAV' opft^ aga xal if VJib AFJ' t^tymvov Si] tov 

AFd at Svo yaviai aC ureo ^AF, AF^ Svo 6pfrafs 

25 laat eiaiv oxeq iatlv aSvvatov. ovx aga i] anb 1 



XVI. Eutoeias in Apollor 



Brl rs B; BT &I/U I). 
.. 4. Sb] 3' BF. 5. aCf^FV. 

fjTtio* P, sed corr. tov 111*15011] tne Siajtfiifov P. \ 

ts'] tij' Fi coiT. ni. 2. 9. ayoiifVTi fvbeCa F et B m. reo^.J 



ELEBiENTOEUM UBER m. 209 

MN = BF. itaque maxima est diametrus A^, et 

Br>ZH. 

Ergo in circulo maxima est diametrus^ ceterarum 
autem proxima quaeque centro remotiore maior est; 
quod erat demonstrandum. 

XVL 

Becta^ quae ad diametrum circuli in termino per- 
pendicularis erigitur, extra circulum cadet^ nec in 
spatium inter rectam et ambitum uUa alia recta inter- 
ponetur, et angulus semicirculi quouis acuto angulo 
rectilineo maior est, reliquus autem minor. 

Sit circulus ABF drcum centrum /1 et diametrum 
AB descriptus. dico^ rectam ad AB in A termino per- 
pendicularem erectam extra circulum cadere. 

ne cadat enim^ sed, si fieri potest, intra cadat ut 
AFy et ducatur JF. quoniam /:IA = /IF, erit etiam 

LJAr^AT^J [I,b],xxerumL^Ar 
rectus est. itaque etiam L AFA 
rectus. ergo trianguli AFA duo 
anguli AAF-^-AFA duobus rectis 
aequales sunt; quod fieri non pot- 
est [1, 17]. itaque recta ad BA in 



12. ndarig B. 13. iaz^v] ^atai in ras. V. 16. AB] (prius) 
inter ^ et B 1 litt. eras. in V. 19. coe] supra m. 2 F. 
AF f. 21. inei] inel ovv p, ante insL add. xa^ m. 2 FV. 
tari iaz^] om. P. ycovia] om. BVp. 22. AFJ iaziv tari P. 

23. JAF] J eras. p. aga] om. B. ri] supra m. 1 F. 
XQiymvov drj zov AFJ at dvo y(oviaL al] F (AF pro AFJ); 
«r aga Theon? (BFVp; a^a et seq. vtto supra m. 2 F). 24. 
dvaiv V. 25. elaiv taai B. iariv] om. p. rov] om. V. 

Euclideg, edd. Heiberg et Menge. 14 





210 ETOIXEIiiN y", 

A STjiisiov Ty BJ irpos og&ag ayoy.ivti ivtog «effsFiS 
rov xvxXov. b^oicas ^17 SEiiofiiv, ort ov8' inl 
xtQKptQtias' ixTog aQa. 

ntitriTio mg ^ ^E' Xdyas dij, oit tig tov (i£t«|ii 
5 trdirov t^s T£ -itfB fu^Etas «ai r^g r@A XEQupegEias 
ExiQa Ev&eta ov aapsftaESEttai. 

El yccQ Svvaxov, xaQEfiTttTtTiTm rag tj 2.A, xwi ^%^ta 
OTtb Tov ^ ffijfiEtou ^wt tijv Zj4 xKS^fTos 1; ^jEf. xal 
iiCEi opftjj ^Oriv ij t»3r6 AH^d, iXaTtav de op&ijs ^. 

10 ujto ^AH, (lEi^av aga ^ A^ r^g ^H. i'6r] di t] ^4 
t^ ^®' t^Ei^mv Kp« ^ ^& T^e z/i/, 57 EAarTov 1 
fisi^ovog' oneg istlv ddvvaTOV. ovx a(fa eis tov f 
Ta^ii Tojroi' t^s ^a EV&Eiccg xal rijg ^iEQKpE^Eiag itifft 
EV&Ela napEfucEOEtTai. 

15 Aeya, oti xal ij (ilv tov ^jfiixvxXiov ymvia ij iMpt- 

E%oy^vn vjlo TE T^s -B^ Bv&Eiag xal T^g V&A XE^t- 
qjEQBiag axdarjg yavlag olsCag Ev&vyQafifiov fiEi^mv 

ietiv, ^ Se ioizi] 57 ItBQlEXOftEVtl VTtO T£ T^g J^®^ ITfipf^ 

(psQEiag xal tijg AE Ev&Eiag «lEKfiijg yaviag o^eia^M 
20 fwSrypajtftou iXdtzmv ioriv, ^ 

£;^ y«p ^UT^ T(S yavla £U#TJypaftf(og fieit/av fihv 
rijg XEQtEXOftivrig vmo te tijg BA EV&Eiag xal i^g 
r&A ntQLtpEQEiag, iXdtrav Si i^g TtE^tE^OfiEVTig 1 
Te T^$r&A jtEQiqiEpEiag xal r^g AE EV&Eiag, Eiq vin 
25 ftftali; TOJTov T^g T£ r&A aE^KpEQsias xal t^s A^ 
EV&Eiag EV^ECa itaQEfiJtEaEitai, ^ttg noi^aEi fiEi^ovi 
filv Tfjg iCEQiBxofiiv^g v%6 ze z^g BA Ev&eiag 
T^g r®A TtEpifpEqEiag vm Ev&Eimv nEQiExofiivrj^ 

1. an' ajcpBS ayOfi^^fj p. 2. ovSf BFp. 4. *ij] om. 

V. 4. r&A] corr. e» FBA m. 2 V. 6. 0«« iiimnltiu 

F; ««(- add. m. S. T. ^raetntinCTai, add, fi m. 1, F, i(} 



ELEMENTORUM LIBER m. 211 

A piincto perpendicularis erecta intra circulum non 
cadet. similiter demonstrabimus^ eam ne in ambitum 
quidem cadere. extra igitur cadet. 

cadat ut AE. dico, in spatium inter rectam AE 
et ambitum r%A aliam rectam interponi non posse. 

nam^ si fieri potest, interponatur ut ZAy et a z/ 
puncto ad Z^ perpendicularis ducatur z/if. et quo- 
niam /. AHA rectus est, et i AAHmmoT recto, erit 
AA> AH [I, 19]. sed AA = AS. ergo A@ > AH, 
mftor maiore; quod fieri non potesi itaque in spa- 
tium inter rectam et ambitum positum alia recta non 
interponetur. 

dico etiam, angulum semicirculi recta BA et arcu 
jr^^compreliensum quouis acuto angulo rectilineo mai- 
orem esse, reliquum autem arcu F&A et recta AE com- 
prebensum quouis acuto angulo rectilineo minorem esse. 

nam si quis erit angulus rectilineus angalo com- 
prebenso recta BA et arcu FSA maior, et idem mi- 
nor angulo comprehenso arcu r@A et recta AE^ in 
spatium inter arcum FSA et rectam AE positum recta 
interponetur, quae angulum efficiat rectis comprehen- 
sum maiorem angulo comprehenso recta BA et arcu 
rSA et alium minorem angulo comprehenso arcu 

in ras. m. 2 V. 9. iXdcaaiv p. 10. JA] AJ F. 11. 

TJ] rijs tp, ^©] © in ras. p. aga] aga %ai p. iXdc- 

cmv pqp. 12. icT^v] om. Bp. 13. rc] om. V. 16. re] 

om. BVp. rSA] r om. B; m. 2 V. 17. o^siag ycoviag 

p. 18. Tj] (alt.) om. P, m. rec. B. ts] om. Bp. 19. o^siag 
yoviag p. o^siag] om. B; m. 2 V. 21. iczi^v P. xig] 

om. p; m. rec. B. 22. rc] om. p. BA] AB ^. 23. iXdc- 
cmv P. 24. TS rijs] om. B; tijg p. 25. ronov] supra m. 1 
P. 26. sv&sia] om. p; m. rec. B. sv^sia, ijcts p. 28. 
VTto] T7IV vno B, vno rs F (ta eras.). vno sv&simv nsQisxo- 
fiivrjv] om. p. nsQisxofisvrjv] -v m. 2 V; nsQislofiivriv P. 

14* 



212 ETOIXEiaN y', 

ilaztova di r^g XE^iexo^dvrjg vno te rijg r®A xbq^- 
tpBQEiag xal rijg AB Ev&iiag. ov TtaffSfiaioTBt Si' 
ovx apa T^s itEQi,t%o^iv7}s yaviag vit6 te t^g BA 
av&siag xal tijs F&A neQttpEQsiag tetai fiEt^ov oista 
B vjio £v&£Lmv XEQiBxoyiiviq, ovS^ n^v ilatTiitv rijg %£qi- 
sxoiiivrjs vJto T£ zijg F&A itspirpe^eias xal zijs AE 
evd-eias- 

77opiffftK. 
'Ex dii TOVTov (pavegov, (iti ^ tij Sia^itgip «ojfv 
10 xvxkov XQog oqQccs an axpag ayo}iivr] iqjdnzetai 
Toti 'xvxkov [xaX oti sv9'Bia xvxXov xa&' kV (lovov ' 
itpdatetai atjfieiov, iaeiS^aep xal ^ xard Svo avt^ 
avii-lidXXovaa ivrog avtov ninzovGa idaixQ"ti]' onsff 
edsi Sst^ai.. 



^Ano toij SoQ^ivtos arjiisiov tow So^ivtoe 
xvxkov itpaatoiiivTiv sv&stav yQap.p,i]v ayaye 

"EaTca tii ^\v tfoftiv aijftBiov to A, o S^ So&eXs 
xiixAos 6 BFz/' SbI Sri dno rov A a^iieiov zov BPjd 
D xvxAou i^antofiivijv evd^Btav ypaiifttjv ayaysti 

Eikritpitto yaff to xivTffov Toij «vxXov ro E, xa^j 
inBiEvxQ^a ^ AE, xal xivTQp (liv ra E Siaatjjfum 
Si rp EA xvxXog yeyQatp&to o AZH, xal 

XVI. itotmna. Simplicina in phya. fol. 12', 



o9m 

:at^M 

ov ^* 

I 



6 tf^ So»bIs 
V xov Br.d^ 
aystv. fl 

' TO £, XK^^I 

Siaot^luctt^M 
tal &na to^.-^I 

. Bp. 6. { 
8, napiffficc] 



ELEMENTORUM LIBER m. 213 

r0j4 et recisLAE. uerum non interponitur recta [u. 
supra]. itaque nullus angulus acutus rectis compre- 
hensus maior erit angulo comprehenso recta BA ei 
arcu r0A nec minor angulo comprehenso arcu F&A 
et recta AE. 

Corollarium. 

Hinc manifestum est, rectam ad diametrum circuli 
in termino perpendicularem erectam circulum contin- 
gere [def. 2].^) — quod erat demonstrandum. 

XVIL 

A dato puncto datum circulum contingentem rec- 
tam lineam ducere. 

Sit datum punctum Ay datus autem circulus BFd, 
oportet igitur a puncto A circulum BF/I contingen- 
tem rectam lineam ducere. 

sumatur enim centrum circuli Ey et ducatur AEy 
et centro E radio autem EA describatur circulus AZH, 



1) Pars altera coroUarii, per se quoqne suspecta, sine du- 
bio a Theone addita est; om. praeter P m. 1 etiam Campanus. 
et re uera corollarium genuinum eodem redit. itaque e uer- 
bis Simplicii concludi nequit, eum partem alteram legisse. 



amBtoci FV. 13. onsQ ^dst dst^ai] postea insert. F. 15. 
i^'] »0-' F; corr. m. 2. 18. iatm — 20. dyayfiv] c/XrJcj)*-» 

ya(f rov Sod^ivtos ytvitXov tov BFJ to So&lv arjfisCov to A, 
xal Itfroo to yiivtQOV tov itvnXov ro £. V; in mg. m. 2: iv 
aXXo} ovtmg ygdtpttat' ieto) to usv do&hv crifisiov to A 6 Sh 
dod^slg nvtiXog o BFd' dst 8ri dno So&svtog crifis^ov tov A tov 
So&svtog TivxXov tov BFJ iq^antofisvriv sv&siav yQafjLfiriv dya- 
ystv^ et ita B, et p {dno tov Sod^ivtog). 19. A] om. tp. 
21. stX^^qj&m — to E] mg. m. 2 V. 22. yiivtQOv tp. 23. 

EA] P in ras. m. 1; F; AE BVp. 



214 STOIXEIfiN t'. 

z/ tfi EAagog 6^9as ^jjdoj ij z/Z, xal iitE%£v%%ioaa,p[ 
al EZ, AB' iBym, ori ano %ov A aijfiiiov tov BFi 
xvxXov iipanTofi^vT] TjXTai t^ AB. 

'Eml yttQ To E xivzQov ietl rmv BF^, AZH 
6 xvxXav, T«ri aga sOtIv tj iilv EA rj; EZ, t] di E^ 
tij EB- Svo Sr\ aC AE, EB Svo zats ZE, EA fffai 
eloiv xal ywviav xoiv^v X£giixovai r^v Jtpog rp E" 
^aatg ftpc ^ AZ ^affsi rjj AB ieij ietiv, xal ro ^EZ, 
Tifiyovov ta EBAtQiydvm 1'6qv ioxiv, xa\ ai Aotnal 

10 yoiviat TKtg ioixaHg yioviaii' l'0'ri apa ij vno EAZ, 
t^ ipjto EBA. OQ&i} Sl 1] vno E^Z' opS'^ aga xal 
^ vno EBA. xai iotiv ij EB ix tov xevtpov ^ dk 
ty StaiiizQO} xov xvxlov wpog oQ&ag ait axQus dyo-, 
(LivTi itpdatstai rov xvxhov • tj AB Squ itpaatstai tov 

15 BFd xvxkov. 

Ano tov KpK So&ivtog eijfiEiov tov A rov So- 
&ivzos xvxXov tov BF^J i^aatoiiivr} Evd^eta yQafifi'^ 
^xtai Tj AB' oTtEQ ISfi noiijdai.. 






20 'Eav xvxKov i^anTTjTai Ttg evd^tta, ajto Sh 
TOti «ivTQOV ixl T^v aqjTjv ixtlevx^fj rtg £v- 
9-£ia, 7} iicilsvx&stGa xd&izos EOTCct inl zijv 
ifpBaToit.ivtjv. 

KvxXov yaQ lou ABF icpaTtriod^Gi tis ev^eta i 

26 -^E xara ro F STjfistov, xal jW^q^&ra ro x^vrpovl 

XVin. SimpUciua in Aristot. de coelo fol. 131". 

1. EJ] AE p. 2. BJr F. 3. xtixiou] ra. 2 poat iip- 
anzoftivT, P, aed add. |3— o. 4. Wl ivxi P. AZH] Z e 
corr. P. 6. AE]EA P. aW V. ZE] EZ B etV 

m. 8. T. cfao] FF, (fa't' uulgo. ntijiiiovaiv P. cijv] 



i 

• M 




ELEMENTORUM LIBER m. 215 

et a ^ ad EA perpendieularis ducatur ^Z, et du- 
cantur EZ, AB. dico, ab A puncto circulum BFjd 
contingentem ductam esse AB. 

nam quoniam E centrum est circulorum BFA, 

AZH, erit EA = £Z, et EA = EB. 
itaque duae rectae AE, EB duabus 
ZE,EA aequales sunt et communem 
angulum comprehendunt eum, qui ad 
E positus est. itaque AZ = AB, et 

A JEZ = EB A, 

et reliqui anguli reliquis angulis aequales [I, 4]. ita- 
que L EAZ = EBA. uerum £ EAZ rectus est. ita- 
que etiam L EBA rectus. et EB radius est; quae 
autem ad diametrum circuli in termino perpendicu- 
laris erigitur, circulum contingit [prop. XVI coroU.]. 
ergo AB circulum BFA contingit. 

Ergo a dato puncto A datum circulum BFA con- 
tingens ducta est recta linea AB] quod oportebat 
fieri. 

xvm. 

Si recta circulum contingit, et a centro ad punc- 
tum contactus ducitur recta, ducta recta ad contin- 
gentem perpendicularis est. 

nam circulum A BF coniingai recta AE in puncto 



om. P. 8. ItfTtV] PF; comp. p; icTi BV JEZ] EJZ 

P. 9. iativ] PF; om. p; iatLBY. 10. rj'] tjj B. EJZ' 
e corr. V; EBA p. 11. t^] tj B; corr. ex tTJg F. EBA' 
e corr. V; EBA iativ F; EJZ y.^ oQ^ii dh ^ vno EJZ' 
om. p. Ttai] om. p. 13. dn a%Qag] om. B. l^, ij Al 
aga Iqxintstat] om. F. 15. BFA P. xvxAov] om. F. 

16. aga dod^ivtog] PF; do&ivtog aqa BVp. 18. 17] m. rec. 
P. 19. «17'] x' F, euan. 24. antiad^a) p. 



216 ETOlXEIfiN v'. 

Toii ABF xvkXov to Z, xal «wo rou Z ^kI ti 
im£*w3:*o ^ Zr'" /^yo, ott ^ Zr xaOETO? ^(rri.w 
TIJV z/£. 

El yuQ jijj, ^i^ra aao tou Z Atl tijw .^£ tttt^&eroi 
5 ^ Zif. 

'Sfffl Qvv 7] vito ZHr ysivia op&i} ieztv, o|j 
a^a ietlv t} vno ZFH' vito Sh t^v ^fi^ova yioviav 
^ lui^tav nXivQU vitoreCvBL' fiei^mv apa 17 ZFt^g ZH' 
teji da ri Zr r^ ZB' fieitmj a^a xal ^ ZB rijs ZH 
10 1} iXtttttiiv tijg (lii^ovog' oniQ iotlv advvtttov. ovx 
«iftt 71 ZH xa&Btog iativ ix\ f^v z/E, b^oliog 3^ 
Ssl^oiiEV, ott ovd' «A^i? Ttg jtAijj' tijg ZF' ^ ZF agtc 
xd&Etog iotiv ixl tijv jd E. 

'Eav aga xvxlov itpanttjTtii rtg iv&sitt, ano Sh 
15 Tov JtEVTpov iitl T^v «93^1/ ffftgfuxfl^s rig fwdEfa, ^ 
ixt^Bvx^Eiaa xd^iTog 'iOTai i^l tfjv itptintoftivriv' 
onsQ 16 si. Sit^ai. 






i 



'Ettv xvxlov iipaxtTjttti tig sii^sta, axo dh 
aO r^s aip^g t§ iipajctofisvti XQog o()ffag [ycavittg^ 
si&sla yQunftrj ax^ij, enl tijg ay_9sisr}g la%atM 
to xivtQov Toti xvxXov. 

Kvxlov y&Q toij ABF itpantie&in ttg EV&sCa 1 

^E xattt t6 r aijfistov, xbI dn,o tov T t^ j^Enifia 

86 opfrnff %#o) 7} FA' kiyco, ott irtl x^g AF ioti 1 

xivtpov tou xvxlov. 

1. To Z] «al foro) 10 Z V. 8. indj eupra m. 2 F. 

7. ZTH] PB, Zrif F; HFZ Vp. Seq. (is/;»v ^oo tif 

ij vno ZHF t^e ">io ZrH ¥ et om. lativ F (in mB. tranaitM 
in V in ras. irant HF et FJf. 9. «o^] m. 2 V, oin. " 

10. ij] pOHlea aiid. V. iXaeniav F. /ot/»] om. p. 
*^] corr. ex SeC m. 2 F. 13. ovii Bp. 13. ti}*] i^e P. ■■ 



JO^ 




ELEMENTORUM LIBER IH. 217 

r, et somatur circuli ABF centrum Z, et a Z ad f 
ducatur ZJT. dico^ ZF ad z/£ perpendicularem esse. 
nam si minus^ a Z ad z/£? perpendicularis duca- 
tur ZH. 

iam quoniam L ZHF rectus est, erit L ZFH acu- 
tns [I; 17]. et sub maiore angulo ^maius latus sub- 
tendit [I, 19]. itaque Zr>ZH. uerum ZF^ZB. 

itaque etiam ZB> ZH, minor 
maiore; quod fieri non potest. 
itaque ZH B,d ^E perpendicu- 
laris non est. similiter demon- 
strabimus^ ne aliam quidem per- 
pendicularem esse praeter ZJl 
itaque ZF a.d ^E perpendicu- 
laris est. 

Ergo si recta circulum contingit, et a centro ad 
punctum contactus ducitur recta, ducta recta ad contin- 
gentem perpendicularis est; quod erat demonstrandum» 

XIX. 

Si recta circulum contingit; et a puncto contactus 
ad contingentem perpendicularis ducitur recta linea^ 
centrum circuli in ducta recta positum est. 

nam circulum ^BF contingat recta ^Ein puncto 
r, et a F ad ^^; perpendicularis ducatur Fji. dico^ 
centrum circuli in -^^JT positum esse. 
» 

14. stpdntSTaL qp, sed corr. 15. enatpi^v p. 16. anTOnivrjv 
p. 18. i^'] H seq. ras. 1 litt. F. 20. Trjg] in ras. m. 1 p. 
ymviag] Theon? (BFVp); om. P. 21. faTai] in ras. <p; 

antecedunt uestigia uocabuli iaTai m. 1. 23. anrea&to P6 
FVp; corr. Simson (Glaeguae 1756. 4P) p. 353. in V «- in ras. 
est. 24. Ante t^ ras. 1 litt. F. 



218 ETOrXEliiN y'. 

Mi) yuQ. alX' bI dwazov^ israi to Z, xal £JEE£ei 

■&to Tj rz. 

'EhbX {ovv\ xvxkov roiJ ABF ^(pdnxaraC rtg eufrj 
ij jdE. ajtb dl tov KEVtpow ml z^v agjiji' Bita^tvmat 
5 17 Zf, ^ zr ciQct xd&srog i6ztv iitl r^v /iE- opS^ 
«pa fflrti' 5) v^to ZFE. ierl Si xal i/ vno AFE opSTj" 
re») ttpa ftfrJv 17 iCTo ZrE Tij ureo AFE if ikarrtav 
Ttj (tsC^ovi' onsp iarlv ddvvarov. ovx apo; lo Z xivtQov 
iorl rov ABF xvxlov. ofioiaig Si] dti^ofitVf ori ovS' 
10 akko Ti nlijv ixl i^s AF. 

'Ettv «Qa xmckov i^axTrjTat rtg ev&sCa, aaro Si 
tijg a^ijg tfj itpanroftivTj nrpoff OQ&ag tv&tta yQafin^ 
dx&fi, i^l rrjg d%&U07is s6r«t ro xbvtqov rou xvxkov' 
OJISQ iSsi det^at. 






l 



'Ev xvxi.a 17 Jtpos tip xivTQ^ yiovCa SiJiX 
eCetv isrl rTjg JtQog rij nsQtipSQsCix, orav rqi 
avtiiv nEQt,fpiQttav ^dsiv &%a>i3iv aC yaviai. 
"Eata xvxlog o ABF, xal rapog filv ip xivrt 
} avtov ycovia iaro} 1] vna BEF, nQog Se tfj ^tptipeQsCe 
17 vno BAF, i%ircieetv di tijv avtijv xtQtqtiQSiav ^d- 
<Stv rijv BF' liya, oti SmXaeCsov ierlv ij ujto BEF 
ycavCa rtjg vno BAT. J 

'E%t%Bv^9tXQa yciQ ^ AE ^t^x*"' ^^^ ^° ^- I 

i 'Ensl ovv (Sri ierlv ij EA rij EB, i0ij xal y<ov£t^ 
il imo EAB rfj vnb ^BA' at Squ imb EAB, EBA 

1. iaxa, To Z] in ras. F. 2. rZl Z e corr. V; ZT p. 

3. ovv\ om. P. «lixiotil -iow in ras. P. 6. ZTE] ZVA 

P. iirwv P. ^r^ P. 6<f»^ - 7. .^rE] mg. m. 1 £_ 

(^ari'» om., ZT^, ..^rj). 7. ZrE] ZEP F ni. 1, ET erwf 

^luoaaiv p. 8. in/v} om- Bp. ZJ Z iriifiEfav V. 





ELEMENTOEUM LIBER m. 219 

ne sit enim, sed, si fieri potest^ sit Z^ et duca- 

tur rz. 

quoniam circulum ^BF contingit recta ^E, et a 
^entro ad punctum contactus ducta est ZF, ZF ad 
jdE perpendicularis est [prop. XVIII]. itaque L ZFE 
rectus esi uerum etiam L AFE rectus. quare 

L zrE = ^rjs, 

minor maiori; quod fieri non potest. 

itaque Z centrum circuli ABF 

non est simiUter demonstrabimus, 

ne aliud quidem ullum punctum 

T extra AF positum centrum esse. 

Ergp si recta circulum contingit, et a puncto con- 

tactus ad ccpitingentem perpendicularis ducitur recta 

linea, centrum circuli in ducta recta positum est; 

quod erat demonstrandum. 

XX. 

In circulo angulus ad centrum positus duplo maior 
est angulo ad ambitum positO; si anguli eundem arcum 
basim habent. 

Sit circulus ABF, et ad centrum eius angulus sit 
BEF, ad ambitum autem BAFj et eundem arcum 
basim habeant BF. dico, esse LBEr=2 BAF, 

ducta enim ^E ad Z educatur. iam quoniam 

EA = EBy 
erit L EAB = EBA [I, 5]. itaque 



di}] corr. ex dsi: m. rec. P. ovSi Bp. 10. in£] om. BFp. 
11. ajctrjtai F m. 1; corr. m. 2. 12. OQ^ag ycaviag Vp. 

16. %p' F. 16. ngog] iv p. 17. iax£v B. 22. BF] FB 
F. BEr yoivia t^s] BV Xiyca oti seq. ras. 3 litt. <p. 24. 
ydg] di F; corr. m. 2. 25. l^ari xcr^] tarj iatl %a£ p. 



220 ETOrXEIflS y\ 1 

yioviat tijs vao EAB SinXasCovg eleiv. lei} Si tj vsro 
BEZ Tcets vno EAB, EBA' xal ij vno BEZ Squ rijg 
vno EAB ioxi dmX^. 8ta ra avra 8^ xal ij wro Z EF 
T^S vxo EAT ia-ci fttTtk^. oAtj «pa ^ t-ito BEroXvis 

'i T^s vno BAF ieri. dixl^. 

KBxXdo&aj Sii ndXtv, Jtal ^ffva tt/pa ymvia ^ -into 
BAF, xttl ixitivx»ttaix ^ JE ixfis^X^o&ca i:tl to H. 
ofioiojs S^ Sti^ofuv, ori StnXij iettv rj vxb HEP yto- 
vltt TTjs vno EAT, rov ^ vao HEB SixX^ iett t^s 

) vith EJB' Xotxii aga fi vno BEF SinX'^ ieri t^, 

vjco B^r. 

'Ev xvxXa Sga ^ XQog ta xivtgct ymvia SiaXaOimv 
ietl r^s Jipog lij xcQiipBQeta, OTttr Ttjv avtiiv xsffi' 
tpiQtittv ^daiv fjjtafffv [ai yaviai}' oat^ iSet SBl^t. 



1 



'Ev xvxXa ai iv ra avra Tfirifiari yoivC&i 
leat dXXijXKis tieiv. 

"Eera xvxAog 6 ABTJ, xal iv to avt^ tfi^fittzi 
Tip BAEd ymviat Seraeav at vno BAid, BE^' 
20 Xeyto, ort af v7to BA/i, BEA ya>viai teat dXX^qXats 
ileiv. 

ECXriqi&a yag tow ABTA xvxlov ro xivxQOv, xal 
eera to Z, xal ^3rsg£Ti;|;*io(j«v at BZ, ZA. 

Kal ixel rj (isv vjto BZA yavia jiqos rp xivrQa 
26 ieriv, ri S\ wro BAJ Jtpog trj jttQtiptQti^, xal iiovei 

1. SmXttaiat tiaiv FV; in SixXaniai nlt. i e co... . , . 
BtnjMaitii, p. 2. ^] om. p. 3. laxtv P. SfiAri tvtl ^ 
4. E^rl in raa. V; corr. ei EZT m. 3 P. tew P. 

BEri htt. BE in ras. F. 6. I«ti» P. 6. tavia iti^tt l^. 
8. jj ijio HEr— 9. eoTi] rag. m, i P. 9. Ei^r] EdVi 
ymvias F. «v] Bopra m. 2 P. HEE] e corr. " '" 



4 




ELEMENTORUM UBER m. 



221 




L EAB + EBA = 2 EAB. 
sed L BEZ = EAB + EBA [I, 32]. quaxe 

LBEZ^2 EAB. 
eadem de causa etiam L ZEF = 2 EAF. itaque 

LBEr=2BAr. 
rursus infringatur recta, et sit 
alius angulus BAF, et ducta AE 
producatur ad H. similiter de- 
monstrabimuS; esse 

LHEr=2EAr, 

quorum L HEB = 2 EAB. ita- 

que LBEr=2BAr. 

Ergo in circulo angulus ad centrum positus duplo 

maior est angulo ad ambitum posito^ si anguli eun- 

dem arcum basim habent; quod erat demonstrandum. 

XXI. 

In circulo anguli in eodem segmento positi inte^r 
se aequales sunt. 

Sit circulus ABFA^ et in eodem 
segmento BAEA anguli sint BAA^ 
BEA. dico, esse L BAA = BEA. 

sumatur enim centrum ciTGxxli ABFAy 
et sit Z, et ducantur BZ, ZA, . 
et quoniam LBZA ad centrum positus est, et 
L BAA ad ambitum, et eundem arcum BFA basim 

iaxt\ comp. supra scr. F. 11. vno] om. B; add. m. rec. 

12. $inXaai(ov] -v supra scr. m. 1 P. 14. cct ymviat] m. reo. 

P; m. 2 V; om. B; in ras. F. 16. %a] euan. F. 16.' «n 

^ , II, III 

om. <p. 19. BAEJ] E supra scr. P. 20. alXrjXaig sCclv 

tcat F m. 1. 24. BZJ] B om. <p, Z e corr. m. 2 V. 25. 
ixovM PB. 




222 rroiXEiJiN y'. 

t^v avtiiv itt^Kpdpsiav paaiv tijv BT^, ^ a^ mto 
BZ^ ytovia dtnkaeiav ierl tijs vno BA,d. Sia tA 
avra Srj ^ v^o BZz/ xal tijs V7th BE/I iazi SmKa- 
aCrov i'0ti aga )j vao BAd Tfj vito BE^. 
b 'Ev kvxX^ aga af iv ra avta tfiiifiati yiavla 
teai aXX^^^aig sleCv ' o^bq SSei. Sst^at. 



Ttov iv tots xvaXoig TitQa7t?.svQ<ov at d 
evavtiov yavCat Svalv 6^&ais i'0at eiaiv. 

10 "Eatfo xvxkos o ABFjd, xal iv avxa tttpaaltv^m 
Satrn 10 ABP^- liyca, oti aC anevavtiov yatvia^ 
Svalv 6Q9ais taat EleCv. 

'EiCElBvx&aeav ai AV, Bz/. 

'EtceI ovv JtavTos rQiymvov at tqels yaniiai SveXv 

16 opdafg !aat eialv, tou ABT «pa tQtymvov ai tQste 
yaviat at xmo FAB, ABF, BFJ Svelv oq&ats itsat 
slaiv. fOij 31 ti fiiv vna FAB tjj imo BAF- iv yuQ 
Tip avTp tft^fiatC dei rp BA^V' ^ Si vito AFB 
TJi ujro A^B' iv yuQ tp avta tfijjftat^ siet tm A^iVB' 

20 olii aga tj vno AJrtats vao BAT, AFB la-ri ietCv. 
xotvi} atfoaxsCa&tn ij vTto ABF' aC aga vjto ABT, 
BAF, AFB xats vjco ABF, AJT iaat slaiv. aXX' 
al vao ABT, BAF, ATB dvalv oQd-ats taat slaCvA 
xal at vao ABF, AATa^a Svelv ogftats laat eleCv 



XSD. Boetina p. 38S, 3? 

3. 171 om. p. BZJ] cort. ei rZ/J m. 1 V. 6. of^S 

Kp tluiv B. avx^] om. Bi supra sor. m. ren. 6. staiv} ota. 
B. 7, xS' P, eroa. 8. d-!tivavtlaiv F, aed corr. 11. Ante 
ymviai add. avrov BVp, P m. rec. 13. ^T. B^l litt. F, 

BJ e corr. P. 14. Infl otiv] xal inei p. 15. tHil Vp. 



•eCv^ 
•eCt^M 

ate 

r, 

M 



ELEMENTORUM LIBER ni. 223 

habent; erit [prop. XX] L^Z,^ = 2 BA^. eadem de 
causa etiam L BZ^ = 2 BE^d. quare 

LBAJ = BEJ. 
Ergo in circulo anguli in eodem segmento positi 
inter se aequales sunt; quod erat demonstrandum. 

XXII. 

In quadrilateris in circulis positis anguli oppositi 
duobus rectis aequales sunt. 

Sit circulus ABF/Jy et in eo quadrilaterum sit 
jiBF^. dicO; angulos eius oppositos duobus rectis 
aequales esse. 

•ducantur AFy BA. iam quoniam cuiusuis trian- 
guli tres anguli duobus rectis aequales sunt [I^ 32], 
trianguli ABF tres anguli rAB + ABr+ BTA 
duobus rectis aequales sunt. sed L FAB = BAF] nam 
in eodem sunt segmento BAAF [prop. XXI], et 

L ArB = AAB', 
nam in eodem sunt segmento AAFB, 
quare L AAr= BAr+ AFB. com- 
munis adiiciatur L ABF. itaque 

^ ABr + BAr+ArB=ABr+AAr. 

uerum ABF -{- BAF -^ AFB duobus rectis aequales 
sunt. quare etiam ABF -^ AAT duobus rectis sunt 

xqiymvov\ om. B. 16. ycoviai dvalv OQ&aig taai sCalv a£ vno 
rAB^ABFj BFA V. 17. Bta^v] euan. F. FABirJB P. 

BJr] BJr F (ante F ras. 1 litt.). 18. siaiv PBF. 
19. ydg supra m. 2 euan. F. sla^] supra m. 2 euan. F; 

slaiv PB. 20. ^<FT^v] PF; comp.p; iatl BV. 21. Post nQoa^ 
%s£a&m in B add. taig $vo ofiov t^ ngog ro A Y,al V %al %oi' 
glg ty p,ia tj ngog to d. 'vno\ (alt.) om. qp, m. rec. B. 

22. ABT\ BT e corr. V. slai B. dXXd P. aU' af — 

23. sUiiv\ om. B. 23. BAT, ATB\ BTA, TAB p. siaiv] 
PF; siai uulgo. 24. aQa] om. BFV. 




224 STOrXEliiN !■'. 

bfioifag 6^ Ssiioitev, oti xal aC vno BA/i, AT 
vCm Svelv dpdctg l^ffai elttCv. 

Tav «Qa iv rolq xvxXoig XBXQanktvQdtv aC 

svavriov yavuci Sv6iv OQ&cits ieai alsiv OK£p iSsi 
i det^ai. 



'E%1 r^g ceur^g ev&tias Svo rfi^iiara xvxXtom 
ofiota xal Kvi6a ov avara&^aetai. ial ra avT^ 

Ei yaQ dwtttov, inl rfi? «ut^s Bv^eia^ r^g j4B 
Svo tfi^iittta XvxXmv ojtota xal aviOa avveOrdra iid 
ra aira ^iQti rd ATB, AAB, xal tft»;^*'^ V -^-T*^» 
xal instevx&tooav a( FB, JB. 

'Eael ovv ofioiov iott, ro AFB rfi^^a ta AAB 

B Tfi^/Aart, o^oia S% TfiJjft«ra xvxAov iOtX tct Sexofisva 

yavCaq fsag, fff]] apa ierlv i/ vno ATB y&vla r^ 

vao Ai^JB 7} ixtog rfj ivtog' oitsp ietlv dSvvatov. 

Ovx aQu in] r^g ttvrijg ev&eiag 6vo tfttjfiara tu 

xKmv ofioitt xal aviBa 0v6rtt9'^6etai inX r« avta (liff^' 

ontQ fSei Set^tti. 



Td inl tacav evd^eimv "(loia rfi^fia 
xXeiv i'aa dXAiiXot.g iotlv. 

"Eataaav yuQ inX idav iv&eiav teov AB, F^ ofioia 
) tfifjnartt xvxXav ta AEB, FZJ' Xiyat, oti fffov iarl 
to AEB rfiijiia ta VZJ Tfi^fiart. 



, 1. «El.n V, 5 
Kf 1 non liquet ii 
PBFp; avma^a 



\ 



!. 2. 


fl«{v] PFpi 


M Bv m 


7. xiixiov f: 8, 


, oudrafl-ijiMTOiy 




l,:l TB aizi 


fifpj]] mg. m. 2 




12. ATB] , 




: r J V 


m. 2. 14. 








ELEMENTORUM LIBER ni. 225 

aequales. similiter demonstrabimuSy etiam 

L BAd + JTB 
daobus rectis aequales esse. 

Ergo in quadrilateris in circulis positis anguli 
oppositi duobus rectis aequales sunt; quod erat de- 
monstrandum. 

xxni. 

In eadem recta duo segmenta circulorum similia 
et inaequalia in eandem partem construi nequeunt. 

nam si fieri potest, in eadem recta AB duo seg- 
menta circulorum similia et inaequalia in ean- 
dem partem construantur AFBj AAB, et edu- 
catur AFA, et ducantur FBy AB, 

iam quoniam segmentum AFB simile est 
segmento AAB^ similia autem segmenta cir- 
culorum sunt^ quae aequales angulos capiunt 
[def. 11], erit L AFB = AAB, exterior interiori; 
quod fieri non potest [I, 16]. 

Ergo in eadem recta duo segmenta circulorum si- 
milia et inaequalia in eandem partem construi neque- 
unt; quod erat demonstrandum. 

XXIV. 

Similia segmenta circulorum in aequalibus rectis 
posita inter se aequalia sunt. 

nam in aequalibus rectis AB, FA similia seg- 
menta circulorum sint AEB, FZA, dico, esse 

AEB = rzj. 

(isag\ seq. spatium 3 litt. F. ^cziv] om. B. y(avCa\ m. 2 

V. 11. 71 ivtog TJ ixTog p. iaxiv] om. p. 24. ydqi] 

supra m. 2 F. FJ] d e corr. m. 1 F. 25. xvxAov qp. 
i^lv P. 

Euolides, edd. Heiberg et Menge. 15 




r 



i:T01XEKiN y'. 



rZJ xal ri&BfiEvov totj (liv A ffijftftoti litl i6 V T-q\ 
Ss AB Ev&iias iwl r^w F^, itpaQiioOei xcel rb B O^- 
HEiov inl 10 z/ e7}(iEiov Sia t6 Hetiv dvai t^v AB 
6 rj; r"^" T^s S% AB ixl rijv FA i(pa^(ioedei!}s iipttp- 
ftdffft xal rb AEB T(iiijia ial to VZ^. eI ywp 57 AB 
iv&Eta ixl Ti}v FA itpaQ(i66Bi., ro di AEB Tftijfta 
iTci zh rZA (lii iq>atf fioOEt, ^iot ivzbs avzai Wfiffefrat 
ij ixTOS ij nttQaXla^Ei og t6 FHA, xal xvxAog kv- 

10 xAoi' xe(ivEi xara JtXtCova etifteta ^ tfwo' onEQ ioziv 
aSvvaTov. ovx uQa ig>aff(io^o(iivi^s t^? AB ev&sias 
inl rijv F^J ovx itpap(i6eti xal tb AEB T(iij(itt inl 
To VZA' ig^aQfioaEt apa, xal teov avTva Iszai. 

Ttt aga inl teav EvS^eidav oftota TfitjfiaTa xvxkejv. 

15 tCtt aAAijAoig ieziv ojceq idEi. dEt^ai. 



KvxXov TfuJfiKTog So^ivTos «QoSavaygailia 
Tov xiixXov, ovnsQ itfTt r^^fta. 

"EOTto t6 So&ev T(i7j(itt xvxXov t6 ABF' SeI: SM 
) Tot' -^BPTfiijfiKTo? jtQoattvayQdjliai tov xvxXov, t 
iezi z(i^ft«. 



1 



L 



1. Eipa^fiaifDfifvoii B, ned corc.j alt. d id rae, V. 3. *a{\M 
om. B. 5. 1^] ^^*" ^i '^"'■'^' "■- ^- IqBnefHniBiuje 3^ (3^ B) 
T^S ^S e^#8/ec i«l i^* r^J BFVp; sed in F ante l^ttQfto- 
otimif legitor: fj 6i AB inl c^v F^; idem in mg. m. 1: tt 8% 
Tqc J^S cc&E^E i^l Tijv r^ iqitifiioadaTis xul 10 >4G ifi^fia 
iirl To rz (i^ ^qioepoffB. 6. rZifJ Zd in raB. F. sf) in 
raa. P. ^ -<B tv^Ha — 8. FZJj om. B. 7. F.^] ^ e 

corr. V m.a. 8. td FZJ] io raa. m. 1 p. iwao^oeti PP. 

^Titi ivTog avTov ntetCxai. ^ ^xtog ^J F; aXXi Tbeon {BF 
Vp). 9. jto!e«lia|ji F. jial MiiiilDe xtiiilov r^fivei] P; k»- 
xlug 9i ifivXov ov Tffivfi Theon (BF?p; in V dc enpra acT. 
m. 1). CampantiB hic proreus aberrat. 10. Svo] P; dvQ, 

aU.a Kal. Tifivii FHJ lov TZ^ xaza nleiova ari^Btit ^ ^M 




ELEMENTORUM LIBER UI. 227 

adplicato enim segmento AEB ad segmentum FZ A et 

posito A puncto in F; recta autem ^B in TAy etiam 

£ punctum in A cadet^ quia AB «= FA, adplicata 

autem recta AB rectae TA etiam segmentum AEB 

in TIsA cadei nam si recta AB cum TA congruet^ 

segmentum autem AEB cum TZA non congruet, 

^ aut intra id cadet aut extra^); aut 

^^ \^ excedet ut THA^ et circulus circu- 

At^ -^Jf lum in pluribus punctis quam duo- 

^^^^-.^ bus secabit; quod fieri non potest 

[prop.X]. itaque recta -<^B cum TA 

congruente fierinon potest^ quin etiam 

segmentum AEB cum TZA congruat. congruet igitur, 

et aequale ei erit [I xoei/. Sw, 8]. 

Ergo similia segmenta circulorum in aequalibus 

rectis posita inter se aequalia sunt; quod erat demon- 

strandum. 

XXV. 

Segmento circuli dato circulum supplere, cuius est 
segmentum. 

Sit datum segmentum circoli ABT, oportet igitur 
segmenti ABT circulum supplere, cuius est segmentum. 

1) Id quod ob prop. XXIII fieri non potest. et hoc ad- 
iicere debnit Enclides; sed non dubito, quin ipse ita scripserit, 
nt praebet cod. P. nam haec ipsa forma imperfecta Theoni 
ansam dedit emendationis parum felicis. 

xa r, H, d Theon (BFVp; xa^ m. 2 V; 6 e corr. p). ^axiv^ 

P; om. BV; ndXi.v F; iffrl naXiv p. 13. xo] xr^v p. rZdj 

rZ litt. in ras. V. Dein in FY add. xftrjfia m. 2. avxo 

y. 14. xa aQoc] aga xd F; ante uQa m. 2 add. xd. xmv 

ftrcoy p. 16. xf. F; corr. m. 2. 18. t6 xfirjfia Fp. 19. 
t6 Sobiv] om. B, m. 2 V. xvxXov xfirjfia B. 21. vd vft^- 
fMC PF. 

16* 



xal 



228 STOIXEiaN y'. 

TiT(iija&co yag ij AT &L%a xaxa ro z/, xal iqx&m 
ano tov d aijfieiov r^ .^F jrpos OQ&ag ^ ^B, xal 
int^ivx&co ^ .4B- ij i^u AB^ ytavia aga rijg 
BA^ ijroi fiEi^mv iatlv ^ l'6r} ^ iXatrmv. 
6 "Eotci nQOZBQOV (iti^tov, xal ein/cffTrarfl) TtQog 
BA tv&Bitf xal ra repos avt^ etjfisi^ ta A rij 
AB^ yavia Haij ij v7to BAE, xal Strjx&ta ^ AB inl 
To E, xaX inBi,Evx%ai ^ EF. iictl ovv tOrj iatlv ^ 
vao ABE yavia rjj vjib BAE, iat] «p« iatl xaX ii 

10 EB EvS^fta tij EA. xal iicEl iar] iatlv ij Ad x\\ ^Va 
«oivTj di ii JE, dvo Si/ at AA, AE Svo tatg r/tj 
jftE taai sialv ExatiQa ExattQif xaX ymvia 17 vjch 
AAE ya>vCa tij vxo F^E iaxiv fujj* op-9-^ yaff ixa- 
xipK' ^aaig UQK ij AE ^kOev xij FE iotiv leij. aXXa 

15 ^ AE TJ) BE iSEix&n fflij- xaX ij BE apa ttj FE 
iativ Cari- al tQEtg «pa ai AE, EB, EF lam «XAij- 
Attig Elaiv 6 Rpa xEvtQip T^ E Siaat^fiaxi. Sh ivl 
tmv AE, EB. EF xvxXog yQa<p6[i£vog ^|« xal Sia 
tmv ).oinmv atjfiEiav xal Satai nQoeavayEyQafi[iivos. 

20 xuxAof ttptt rftijfiarog liofffWos XQoeavayiyQumai 
6 xtwAos. xaX diji.ov, cog to ABF tfi^fia ^Aottoi 
iaxiv Tjfiixvxliov Sia x6 t6 E xevxqov ixtag avTi 
tvyxdvetv, 

'Ofioiag [Se] xav y ^ vjtb ABjJ yavia fffi; r^ imh 

86 BAA, rijs AA tOtjg yevOfiBVtjg ExaxiQa xcov Bz/, ^V 
al XQElg al JA, /JB, AF i^aai aXi.i^Xatg iaovxai, 

1, yui)] om. p. diijlffoi F. .3. aQU ymvin 

Tp p. 7. Post JB eraB. ho^ V. 8. loriv] comp. Hupra I 
m. 2. 9. iKo ABE — 10. ('oij iinlv 71] om. B. BAE] »" 
in rns. p. iotiv F. 10. EB] BE P. t§] ev&ila ™ P. 
EA] P, F m. 1, V m. l; AE F m. 2, V m. 2, p. 11. diit} 
(alt.) 3vi,i V. 14. piaig] Pj Ka! faats BVp; in F x*!- supn 



i 



nroit^H 
uwo 



ELEMENTORUM LIBER III. 229 

nam .^F in duas partes aequales secetur in ^^ et 
a A puncto ad AT perpendicularis ducatur z/E, et 
ducatur Ah. ergo L ABJ aut maior est angulo BA/^ 
aut aequalis aut minor. 

Sit prius maior^ et ad rectam BA ei punctum 
eius A construatur iBAE = ABA [I, 23 J, et edu- 
catur ABB.dEy et ducatur EF, iam quoniam 

LABE^BAE, 
erit etiam EB = EA [I, 6]. et quoniam 

.d-2^jtAA = AF^ ei AE communis est, duae 
rectae AA, AE duabus rA, AE aequales 
T sunt altera alteri; et i AAE = TAE'^ 

nam uterque rectus est itaque AE=TE [I;4]. uerum 
demonstratum est^ esse AE = BE. quare etiam BE 
=TE. itaque tres recidie AEyEB,ET inter se aequales 
sunt. ergo circulus centro E, radio autem qualibet 
rectarum AE^ EB, ET descriptus etiam per reliqua 
puncta ibit et erit suppletus [prop. IX]. ergo dato 
segmento circuli suppletus est circulus; et adparet^ 
segmentum .^BFminus esse semicirculo^ quia centrum 
E extra id positum est. 

Similiter si i ABA = BAAy tres rectae AA^ 
z/B, AT inter se aequales erunt, cum AA = B^ 

■ ■ 

8cr. aXXa] P, V m. 1; aXX' F; aXXa -Aai Bp, Vm. 2. 16. 
A%\ ABY. BEl (prius) bis F (semel m. 2). 16. tQr\ lexlv 
p. E^ P. aUiJXofiff] om. V. 18. %ai\ om. P. 19. 

TTpotfttVttypaqpofifvos F; mg. m. 1: y^. nQoaavttysYQocfifiivog, 

20. xvrtXov] 6 %v%Xog, nvnXov P. In B mg. lin. 5: ^XarToy 
^fii%v%Xiov, lin. 24: ri(U%v%Xt0Vf p. 230,3: fisr^ov ritii%v%X£ov. 

21. iXatxov] mg. m. 1 P. 22. to E] in ras. p; E P m. 1, 
B. 24. $i] in ras. V; om. P. %av 17] %al idv P; %av seq. 
i in spatio 4 litt. tp, ABJ] corr. ei ABF m. 1 P; Bd in 
ras. V. tari ^ P. 26. z^ri ^ in ras. p. 26. x^Big] P 
m. 1, F, y seq. ras.; xffBig aQa fip, P m. rec. 




230 .STOrXEIflN y'. 

xal iaxttt xo A xsvTQOv tow itpoaavaxsal7}pmp4 
xvxXav, xal drjkadi] lazcci to ABF ■tjiiixvxJLtov. 

'Eav de 7} vjtb AB^ lldTrav y rijs vnb BA^, ' 
xal <JvaTti6iB[is9a npbg rri BA Ev^eia xal t^ nQog 
6 avrfj <ft}(i8ia tw A rjj vno AB^ yavia fffjji/, ivxos 
xov ABF T[iti(ittTog jKaBlTai to xbvxqov ial r^g ^B, 
xal iaxtti dTjXadii xb ABV Tfi^jiK ^bI^ov i}(uxvxliovr^^ 
Kvxiov aga rjtijjiaros Soifivxos itQoaavayiyQanxtuu 
xustAos" onsQ bSbi sroi^OKt. 



'Bv tofg iaoig xTJxJ.ots a[ laai yioviai 
teav %£QitfiB(feimv psfiijxaatv, idv ze npog r 
xivTQois idv XB XQbg rais nBpi^sQsiais i 

"Eataaav l'aot xvx.kot ol ABF, AEZ xal iv 
toie taat ycoviai iatmaav ngbs [ihv rots xivtpois aC 
vxb BHr, E0Z, aQos Si xatg mQiqnEQSiaig at vab 
BAT, EJZ- Uysa, oxt lai\ iaxXv ^ B K F nBfft<pi(feia 
xtj EAZ nsQitpspsitt. 
' 'EaB^BVxQ^etaav yaff a£ BF, EZ. 

Kal insl Faot sialv oC ABF, JEZ xuxAot, teat^ 
sialv aC ix xmv xEvxQtav dvo dij aC BH, HF Sv<r 
xatg E®, &Z tOaf xal ytovia ij nQos xa H ytovi^ 



8. ^B-Jj ^cq, apatium 3 litt. (p. 4. avyeTtioa>iu&a 

avacjiooitt&tt Bh'Vp; corr.B tn.tec ngos orrn] P; J TheoB 
(BFVu). 5. Tffl A] Pi om. Theon (BFVp). yiaviuv FVp.l 

fariv) corr. ei hTi m. rec. B. 6. t3B] B in raa. p, Dein 
udd, ag ta E mg. m. 2 P; as to 6> aapra, m, lec, B, mgp. m. 
2 V. 7. iiinKvnXiov] neq. spat. 2 Utt. ip. 8, Kunioi'] om, 

Bp. cftiiftaTOE aga Bp. irpoo- Oiu, fiVp. 9. HiiHilag 



£L£M£NTORUM LIBER m. 



231 




P, 6] ei ji^ = JF] et z/ centrum erit circuli sup- 
pleti, ei ABF semicirculas erit. 

Sin L ABJ < BA/I, et ad 
^A rectam BA ei puuctum eius A 
construimus angulum aequalem 
angulo ABJ [I, 23] ^ centrum 
Jjp in recta AB intra segmentum 
w^^Fcadet, et segmentum ABF 
maius erit semicirculo. 

Ergo segmento circuli dato suppletus est circulus; 
quod oportebat fierL 

XXVI. 

In aequalibus circulis aequales anguli in aequalibus 
arcubus consistunt, siue ad centra siue ad ambitus 
consistunt. 

Sint aequales circuli ABFy 

^EZy et in iis aequales an- 

guli sint ad centra BHFj 

EGZ, ad ambitus autemB^F, 

EAZ. dico, aequales esse 

arcus BKr, EAZ. 

ducantur enim BF, EZ. et quoniam aequales sunt 

circuli ABFj AEZ, etiam radii aequales sunt. ergo 

duae rectae Bif, HF duabus E®, SZ aequales sunt; 




ovniq iaxt ro tfiijfioc V. noirjcoci] Ssi^ai PF; in F mg. m. 1: 
yQ. noiricai. 10. x?'] sic tp. 13. ciaiv B. 14. ^c^ijxviatl 
postea add. m. 1 F; m. rec. P. 15. iaxtocav yuQ P. xal 
nQog fisv toig TiivtQoig Caat ytavCat iotoaaav P. 17. BHF] 

post raa. 1 Utt. F. 22. BH] HB BVp. $vo] (alt.) Svat 

V; Svaiv p. 23. E9] 9E V, corr. m. 2. taai] P, P m. 1; 
icat slal BYp, F m. 2. xA] to B. 



232 



ETOISEUiN y'. 



T^ mpos ra & leti ■ fiaSig ttpa ij BF fiaOEi. zfi EZ 
icztv i"tfij. xal iitEi Harj iaxlv ij wpog ra A ycovia r^ 
apog T^ A, ofioiov Egtt EBtt ro BArrnijfia rp E^JZ 
tff^fiutf xaC eIblv i'zl Camv ev&aimv [Ttow Br,EZ]' 
3 Ta iJI ^Jtt leav Bv&umv ofioia T[i^{i.ata xvxkav ttsa 
K^AijAois iotiv teov aga lo BAF Tfi^fta rra EjdZ. 
itftt dl xal oAog 6 ABVxvxlog oAoj rw ^£ZxvxiI,pJ 
fffos" AoiH^ ap« ^ BKF itegtqt^peta rfj EAZ xtQi-H 
ipsQEttt iattv torj. 
10 'Ev aga rofg taoig xvxAots a^ /iJai j^rau^Ri ind fflo 
^Epi^ifpfimi' ^E^^^xatfiv, ittv ts n;pog rofi; 3CEWpo($ ^cfifi 
t£ npos tatg XEgnpsQeiag mai ^E^tjxvtaf ojcbq iSet 
dEtlat. 



'Ev lofff 



ipBQEIl 

iav X. 



zolg ■ 



jcAotg at i-iiX tatov nBpL-4 
oviat taat dXkijkaig eieiVfM 
vrpotg iav te Jtpog 
7CEQiipE0£ittig rofli ^BJi^rjxvtat. 

'Ev yag taoig xvxloi.g tolg ABF, AEZ inl teatA 
20 jiEQKpEpEimv riav BF, EZ jrpog (liv totg H, ® xiv- 
rpotg yaviat ^E^rjxitmaav aC v7t6 BHF, E&Z, npos 
Si tatg ntQtfpEQBCatg aC viio BAF, EAZ- kiym, oti 
jj [liv vxo BHV yavia ty vxo E&Z iattv tari, t] i 
vno Bjr trj imo EAZ iaziv ta^. 



XXVn. Boetina p, 368, 5. 



. t^] 10 B. i-oij] PV, Fj 



. 1; iaxtv terj Bp: fn] ivf 
(prias) T(i B. _ lexlv P. 



i. im* Br, EZ] mg. m. rec, 1 
m. 1 P. e, BJr] litt. BJ 

1 litt. F. EJZ] niGtat. in J 
JEZ] E inflert. m, 1 F; EJZ Bp; JEZ tag. m. a V. 



ELEMENTORUM LIBER m. 233 

et angulus ad H positus angulo ad posito aequalis 
est. itaque Br=EZ [1,4]. et quoniam angulus ad 
A positus angulo ad z/ posito aequalis est, segmenttun 
BjiF segmento E^Z simile est [def. 11]. et in 
aequalibus rectis posita sunt. segmenta autem si- 
milia in aequalibus rectis posita inter se aequalia 
sunt [prop. XXIV]. itaque BAF ^ EJZ. uerum 
etiam totus circulus ABF toti circulo AEZ aequalis 
est. quare qui relinquitur arcus BKF arcui EAZ 
aequalis est. 

Ergo in aequalibus circulis aequales anguli in ae- 
qualibus arcubus consistunt, siue ad centra siue ad 
ambitus consistunt; quod erat demonstrandum. 

XXVII. 

In aequalibus circulis anguli in aequalibus arcubus 
consistentes inter se aequales sunt, siue ad centra 
siue ad ambitus consistunt. 
.A^^^-.^ ^ nam in aequalibus circulis 

^f\ \. ABF, AEZ in aequalibus 
arcubus BFy EZ ad centra 
H, S anguli consistant BHF, 
E&Zy ad ambitus autem 
BAr, EAZ, dico; esse L BHr= E&Z, et 

LBAr= EAZ. 




nvnXat] in ras. m. 2 Y. 8. x^] iativ Hcri x^ P. EAZ] litt. 
AZ in ras. V. 9. iaxLv tarj] om. P. 10. *Ev] inter s et v 
1 litt. eras. V. 12. <Saiv F. 14. x^'] sic 9. 18. (oaiv 

P. 19. %al inC F. 23. ytovCa] P; om. Theon (BFVp). 

EGZ] corr. ex EBZ m. rec P; BHr 9. 24. iaxiv tcri] P; 
om. Theon (BFVp). 



234 2T0IXEIHN y\ \ 

El yaq aviaos ^ortv ij vno BHF rij vxb E&Z, 
[lia avTtov fisi^av iativ. fOio) {lEi^av ij imb BHF, 
nal avvBaTaxta itQog tfj RH Bv&tCa xal Tto jrpog avrji 
ftjfiaiat Tp H tri vao E&Z ycavia iar) ij vao BHK- 
5 a[ di iaai ymviai inl iatav JiEQUpBqEiav ^e^^xaOtv, 
otav agbs rofs xfVTpotg coaiv l'ari ap« i} BK icsffi- 
«pspata T^ EZ jreptgsEpj^a. a},i.a ^ EZ rrj BF ioztv 
lari- xal 1} BK aga t^ BT iativ fOTj ^ iXatTmv tf} 
fiei^ovt.' ojTfp iarlv advvatov. ovx aga aviaog iotiv 
10 Ji vnb BHV yavia rfj VTtb E0Z- fijjj aga. xaC iati 
t^s (i.lv v«b BHF Tj(iiaei.a ij ;rpos rto A, t^s ^^ wro 
E&Z ijftiaiiu ij repog TtJ ^" fiJ»; apa Kai ^ ffpos Tji>« 
-i^ y<avia tfj jrpog t(o z/, 

'£v apa ToEg taoig xvxXois at inl fotav tcsqi^s* 
15 pEifDi' fie^ijxvtat yatviai faai ai.l^Kais tlaiv, idv 
jigbg Tofs xivtgois idv ie npbg tatg nsQifpEQECatq taei J 
fiefirixvtaf ojtEQ Idsi SEt^ai. 
jfij'. 

'Ev Tois taoig xvxkoig at i'aai Bv&etai Haccff*. 

ao negtipEgeiag dipaigovat t^v fiiv lisi^ova tfj fte^^A 

£owi T^i/ Si iKdttova tfj iA.dTtovi. 

"Eatetaav Caoi xt;»Xoc of ^BT, /JEZ, xai iv tol^^ 
xvxloig taat ev&etat Satiaaav at AB. AE tag fiil^ 
AVB, AZE 7CEQt<pEQEias fiEi^ovas afpaiQovaai tag i 

1. li ydp atmds iaziv it v-kq BJjrTj/ ujio E0Z] PF; om."! 
V; tt (iiv ovv i, ijto BHP fm, lotl (^arl'»' B) t.} *j.o E»Z, 
qiKWpo»', oti «al 1] ■uno Bv<r Anj iavl {Iniv B, om.V) Bp vxo 
EJZ- tl Si Do Bpi ia V eadem mg, m, 2 exMpti» i? *i o5, 
quae in textu aunt m. 1 (d S' oS). yp. lutl ouTa;- tf fii* — 
SAV iB ^"" E^Z lifTj tori^v t^ *i ouj ^Ca ovrii' ^eCimv ^ 
iino BHr, xal ovvEaiBTia Moi xa4f£^E tos Iv im xci^fv^ mg. 
m. rec. P. Campanus cum PF concordat. 2. fif/gaii' iatii] 
Bp! iaxi lititav FV; ^ei-ifB* fflia. P. tflrai (in^^inf] om. Y, 



ELEMENTOBUM LIBEfi UI. 235 

nam si L BHF aDgulo E&Z inaeqoalis est, alter- 
uter eorum maior est. sit maior L BHFj et ad rectam 
BH et punctum eius H angulo ESZ aequalis con- 
struatur BHK [I^ 23]. et aequales anguli in aequa- 
libus arcubus consistunt^ si ad centra sunt positi 
[prop. XXVI]. ergo arc. JJjSl = EZ. sed £Z — JJP. 
quare etiam BK ^ BF^ minor maiori; quod fieri non 
potest. itaque L BHF angulo ESZ inaequalis non 
est; aequalis igitur. et angulus ad A positus dimidius 
est anguli BHF, angulus autem ad ^ positus dimi- 
dius anguli EBZ [prop. XX]. itaque angulus ad A 
positus angulo ad jd posito aequalis est. 

Ergo in aequalibus circulis anguli in aequalibus 
arcubus consistentes inter se aequales sunt; siue ad 
eentra siue ad ambitus consistunt; quod erat demon- 
strandum. 

xxvm. 

In aequalibus circulis aequales rectae aequales ar- 
cus abscindunt maiorem maiori^ minorem autem mi- 
nori. 

Sint aequales circuli ABF, ^EZ, et in circulis 
aequales rectae sint AB, AE, arcus AFB, AZE 

add. rxj, cui nanc nihil respondet. 3. svd^sia] om. p; mg. 

m. 2 V. 4. E©Z] in ras. m. 2 V. 7. uXi' Bp. tai^ 

iaxC Yq>. 8. Sr x^ BJt B m. 1, Fp, V m. 1. 10, iativ 

P. 12. tani uQa %cil — 13. tc5 J\ om. F. 13. tc5] x6 B. 
14. Iv apa] e corr. m. 2 V. 15. ^B^ri^vCai ymvlai\ q), seq. 

«i m. 1; in P ycavLai supra Bcr. m. 1. i6. ^ePriiiviai. iaiv P. 
18. X' F. 19. taai] taai tp (non F). 20. aqxuQOvaiv P, 
atpBQOvat q>. 21. ilaaaova xij iXdaaovi V. 22. toig 'nvnXoig'} 
P; avtoig Theon (BFVp). 23. AB, dE] P; BF, EZ Theon 
(BFVp). 24. ArB] P, F m. 1; BAF BVp, F m. 2. 

JZE] P; EJZ Bp, V e corr. m. 2; dZ inter duas ras. F. 
iq^BQOvaai P; q>SQOvaat V, corr. m. 2. 



ETOlXEUiN y\ 



Ev Ar^a^ 



AHB^ A&E ^iwiTovag ■ ^eyto, oti i\ [liv 

grow jtBQi.fpBQua tori ierl zij AZE (tsi^ovi. nequptQBia, 

7} Si AHB ilaztav irepi^^^fta i^ A&E. 

ElXrifp&a j-ftp tK xivTQa rmv xvxlav xa K, A, 
5 iTcs^Evx&ettSav at AK, KB, AA, AE. 

Kal EHfi teot xvxAot sieiv, fffat elel xal ai ix 
xivTQaV Svo d^ aC AK, KB Svel tatg A A, 
[eai tteCv xal fiaeis ^ AB ^aesi rfj AE ler}' ye 
aga fj vito AKB yatvta vri vxo A AE ffftj ietiv, al 

10 teai yiaviai ixl ieav Jtfpi^sQtitov ^t^i^xaeiv, 
jtf/os tots XEvtQOig aStv Coi] aga r] AHB ^egiipdQSt 
tfl A&E. ietl 6i xal o^og 6 ABF xvxkog oktp ri 
AEZ xvxla teog' xal loiJiii apa r{ AFB neqitfiiQBta 
Xotstfj ffj /JZ.E ittqitpEQEia tei) ietlv. 

16 'Ev «pR TOis leoiq xvxkois "-^ tea.1 ev^tiai f« 
jiEQtcpEQBias arpatQOvet tip; (liv fiEi^ova r^ [ititf 
tijv Si ikattova tij ildttovt ' ojtiQ iSsi Sst^ai. 



] 



1. AHB] P; BHr BVp, F in raa. ^ȣ1 P; 

BFVp. .jrB] PF; B^r BVp. S, ^oi^j om. B. 
— 3. r^] om. B; r^ EiJZ fie^OFi atgiqieQgia i] Si d HB (( 
ilaTTBiv jttfiiipidtia teri ij mg. m. ree, JZE] PF; BiJZ 

BVpqj. 3. jiHB] P (B?); BHF Vp, F in raa. iir^ 15 

BFp, tati iatl tij V. ^&E] P; E&Z kdTtovi Bp; EflZ 

^liitDn TteeiatQtiif V, F (ESZ in ras.). &. ^mjiu^t^ooa* 

rp. AK\t; K& BV. P in ras , p (X in ras). .KB] P; 

Kr BVp, F in ras. 'iAVP; AE V e corr. m. 2, F in rss.; 
EA Bp. ^EJ P; .-IZ BVp, F in taa. 6. taat ilo{] m. 

rflc. P. aC] supra m. 1 P, m. S B. 7. .^X, XB] P; BX, 

Kr flVp, F in ras. 3viri] Svo F, corr. m. 2; Svaiv p. 
^.4, ,iE] P (J-1 corr. cx ^^ m. rec); E^, -iZ BYp, F in 
raB. 8. taai eleCr] PF; i'oa< (^a^V et add. m. 2 Bp. ^B] P; 
BT BFVp, dE] P; EZ BVpv. 9. «"<>] om. Bj 

AKB] P; BSr BVp, F in raa. ^AE] P; E^Z BV 

ia ras. 11. AHB] BHV V, in rb. Fpj iixi, BHT B,^ 

del. nipnpiecia] om, B; in ras. p. " " " 

p, pOBt raB. V, in raa. F; v«i ESZ, del. 




ELEMENTORUM LIBER ni, 237 

maiores abscindenteS; AHB, ^®E autem minores. 
dico, esse arc. AFB = ^ZE, AHB = ^&E. 





sumantnr enim centra circulorum K^ A, et du- 
cantur AK, KB, /JA, AE. et quoniam aequales cir- 
culi sunt, etiam radii aequales sunt [def. 1]. itaque 
duae rectae AK, KB duabus ^A, AE aequales sunt; 
et AB = jdE. itaque L AKB = AAE [I, 8]. sed 
aequales auguli in aequalibus arcubus consistunt, si 
ad centra sunt positi [prop. XXYI]. itaque arc. 

AHB = ^&E. 
uerum etiam totus circulus ABF toti circulo AEZ 
aequalis est. quare etiam qui relinquitur arcus AFB 
reliquo arcui AZE aequalis est. 

Ergo in aequalibus circulis aequales rectae aequales 
arcus abscindunt maiorem maiori minorem autem mi- 
nori; quod erat demonstrandum. 



ns^upBQBia B. iativ P. ABF] m ras. F. 13. JEZ] E 
supra m. 1 F; EZz^ P. taos] insert. m. 2 F. %ai] PF; 

om. BVp. ATB] F; ABF F; BAF BY^. ns^itpiQSioc] 

om. Y. 14. Xom^ tij] in mg. transit, antecedit tarj m spatio 
plurimn litt. y. JZE] scnpsi; JEZ PF; EJZ BVp. 
15. lat taai svd^stai] in ras. F. 16. atpaiQovaiv F, -9«- e 

corr. V m. 2. (isiiovi] post lac. 8 litt. in mg. transiens q>. 



'Ev toig i'6ois xvxlois Tag [eag neQi.g>sp£tag 
leai tv&stai vaoTiivovffiv. 

"Edtmeav fffot xvxXoi ot ABF, AET,, xal iv av- 
6 rofff iffai. XiQKp^paiai a^rit^iJ^n^roflKV aC BHF, E&Z, 
xal ^nr^f ifjjftwffKi' aC B V, EZ sv^stat ' i.eyin, oti fffij 
ifftlv 1) Br trj EZ. 

EiX^qi&ca yag tu XEVtpa tcov xvxXtov, xal iera 
r« K, A, xal iTtsitvx&iaeav «f BK, KF, EA, AZ. 
Kal iiiBl lev} ifftlv ^ BHF msgitpiQfia tij E&Z 
X£Qi<p8peia, iffTj ^azl xal yavia rj vxo BKT ty vnb 
EAZ. xal iitKl ieoi slelv ot ABF, AEZ xvxf.oi. teat 
aiel xttl at ix tav xivtQmv Svo dii aL BK, KF Svol 
ttttg EA, AZ teat tieiv xtel yvaviag teag atptixeveiv 
5 /3»'ff(j «pK 7) Br pdeii. tfj EZ Tffij ieriv. 

Ev ttQtt tot^ 1'eoig xvxXoig tas l'ettg asQttpEQeiag 
taai evQsiai vMoziivovsiv oaep sSei Stt^ai. 



Ttiv So&eteav vteQifpiQeii 



XXX. Proolus p. 272, 15. Boetiuf 



\ 



1. !«■ F; corr. m. 2. 2. vno xas FV. 

'8'£ra(j fvd-fiQi V, ^tiai F, quod iu tv^siai corrigere coData 
Gst m. 2. vnozelvavviv'] vTcateivovuiv Heai V; VTtoielvovat 

(in ras. m. 2, punctia del.) iv&Blai «no {nig. m. 2), dein «^- 
vovaiv m. 1 F. 4, ^aai] sapra, m. 2 V. Iv] ajiHX^ip&iaea* 
iv V. 6. foat niQiipe- in mg. m. 2 post 7 Utt. enan. F. 

d^iilijfQtaoav] om. V. 6. BT, EZ tv»eiai'\ e corr. m. 2 P. 
7. Br] Sr ev&tia BVp; (u&fiir in P add. m. rec, in F in 
mg. m. 1. EZ iuffK^a V ro. 2. 8. iHijcp&ra— 9. _AZ] om, 
V. filriip9aaav p. mdI i'ffT(d] P, loioi F (sed mixloJi' re- 
nouatum); om. BVp. 10. *al iitei] Uel Bp; el f ap V m.l. 
(Xfl ytfp V m, 2. 11. laxCv P. BAr] K e corr. m. 2 V. 



ELEMENTORUM LIBER m. 239 

XXIX. 

In aequalibus circulis sub aequalibus arcubus ae- 
quales rectae subtendunt. 

Sint aequales circuli ABF^ jdEZ, et in iis ae- 
quales arcus abscindantur BHF, EGZ^ et ducantur 
rectae JJF, EZ, dico, esse BT^EZ. 





sumantur enim centra circulorum et sint K, Ay 
et ducantur BK^ KT^ EA, AZ, et quoniam arc. 

BHF^ E&Z, 
eTiteiiBmLBKr=EAZ [prop. XXVII]. et quoniam 
circXLli ABFy^EZ aequales sunt, etiam radii aequales 
sunt [def. 1]. itaque duae rectae BK, iJLFduabus EA, 
AZ aequales sunt; et aequales angulos comprehen- 
dunt. itaque BF = EZ [I, 4]. 

Ergo in aequalibus circulis sub aequalibus arcu- 
bus aequales rectae subtendunt; quod erat demon- 
strandum. 

XXX. 

Datum arcum in duas partes aequales secare. 



13. Mv PF. af] om. P. ix] om. p.* U. Bloiv] PBF; 
ilai Vp. iaag ymvicig Bp. nBQiB%ov6iv^ PB, nBQiB%ovct 

pqp, nBQivpBQOvai.v V. 16. vno xdg BFVp. 17. al taoci. V. 
onsQ idsi. dsiiat] m. 2 F. 18. V] non liquet F. 



240 STODCEIilN y', 

"EaTCO rj SoS-stea ^EQirpeQeia j} AJB' Ssl 8i) ri 
j4JB msQttpsQtiav Siicc xE^etv. 

'EJKEi,svxQG} ^ AB, xk! tBT(i.rj6&fa SCya » 
x«i aito xov r arjiisiov rj/ AB sv&tia TtQOg 6q&0!\ 
5 jjx^^ V ^^) ""'^ ^Jtf^fv;j9-(affav af A^, ^B. 

Kal iml Htfrj istlv ?; ^F ry FB, xoivii Si 
rA, dvo djj al Ar, rj SvaX ratg BF, r.d taai 
tiaiv xal yavia ^ ujto AF^ yaviq, rtj vjcb BFd 
iai}' oQ&ij yap ixazBQa' ^aaig «QCc ij AA ^ttOeL t^ 
10 -^B fffij iativ. aC di tOai evQ-stat. teag TtsgigisQsias 
a(pat(fov0i. zijv (liv (iti^ova r^ fisi^ovi rijv Si ildztova 
tjj ikttztovf xaC ieriv ixatepa tmv AA, ^B xeqi- 
ipeQEmv ilttttav '^fiixvxi.iov terj a^a fj A^ refipi 
cpsQEia tij ^B XEQitpEQsia. 
15 'H apa So&Etaa iteQiq^sQEitt Sixtt Terntjttti xata 
jd 0i][iEtav oTieQ ISei aoi^eai. 



Xa. 
'Ev xvxXm ii filv iv ta ^(tixvxkim ymvi 
6q&'^ ietiv, ^ Si iv ta fni^ovt Tfnj/iart iXu- 
20 zav opff^s, 7j Ss iv rp ilattovi tfi^^nati ftsi- 
§or op&^g" xal Sti, tj ftiv tou (ieC^ovos nii^^iia 
tog ymvia (tsi^av iatlv dp^^^g, ^ Sl rov iXdr- 
tovog r^^^KTOs ytavia ildtttav 6qS 



"M 



X5XI. [Euclid.] opt. 47 (Studien p. 132). Alexi 
Aphrod. m melapli. p, 318. SimplioiuB iu phy9. fol. 14'' 
lop, in aaal. 11 fol. 86". BoGtiiiB p. 888, 10, 

1. AJB] litt. JB in raa. V; AB corr. ex AT P. 
ABJ Bp; AB P. ■ 3. B^x"] ^ AB Sifa V. 6. F^ 



i 



^ 



ELEMENTORUM LIBER m. 241 

Sit datas arcus AJB. oportet igitur arcum AJB 
in duas partes aequales secare. 

/1 




A T B 

ducatur ^^ et in duas partes aequales secetur in 
r [I, 10] y et a puncto F ad rectam AB perpendicu- 
laris ducatur FA^ et ducantur AA^ ^B. et quoniam 
AF = FBj et communis est FAj duae rectae AF^ 
r^l duabus BTj TA aequales sunt; et 

L ArA^BTA\ 
nam uterque rectus esi ilaque AA = AB p[; 4]. 
uerum aequales rectae aequales arcus abscindunt ma- 
iorem maiori minorem autem minori [prop. XXYin]. 
et uterque arcus AA, AB minor est semicirculo. 
itaque arc. AA = AB, 

Ergo datus arcus in duas partes aequales sectus 
est in puncto z^; quod oportebat fieri. 

XXXI. 

In circulo angulus in semicirculo positus rectus 
est; qui autem in segmento maiore positus est^ minor 
rectO; qui autem iu segmento minore positus est^ 
maior recto^ et praeterea angulus segmenti maioris 
maior est recto^ minoris autem segmenti angulus minor 
recto. 

ras. m. 1 P. 12. iXdxovi^ P. ixatiQOiv (p, rmv^TOv 9. 
JB] om. F. 14. JB] ia ras. V. nsQLfpsQsia] om. V, nsqi- 
(psQStav 97. 15. ij] in ras. V. 16. notijcai.] dsi^ai P. 
17. Xy' F. ^ 18. iv] post ras. 1 litt. V. 22. ymvia] m. 2 

V. 23. OQdijg] PF; satlv o^^g Bp; OQdijg htiv V. 

Euolides, edd. Heiberg et Menge. 16 




242 ETOIXEIHN v'. 

"Earta xvxlog 6 ABV^, diaiiSTQos 6h avtov ietta 

7] BF, xdvTQOv 6i t6 E, xal ix£%tvx&a>*sitv at BA, 

AT. AJ, ^r- U-yca, ozt, rj fihv iv ta BAF fifii- 

xvxXigj ymvia 7] V7i6 BAV opftjj iaTiv, fi Si iv rp 

& ABF ^siiflvi rou iiynxvxXiov ift^^ari ymvia f) vjto 

ABF iXoLTTfov iatlv opS^s, »/ 3i iv ta A ^} F Hatrovt 

Toii rjnixvxXiov Tfiij^oirt ymvia {] ^o AjdF (leitam 

iozlv 6p&7Js. M 

'EhcB^evx&at vi AE, xaX Si'^x%in i\ BA ixX to Z. * 

10 KttX i%d fffj; ^ffrlv ri BE t^ EA, tatj iarl xal 

■ytavia ^ vito ABE tfi fnto BAE. ndXtv, iael fffjj 

iar\v ii FE tTj EA, fiHj ietX xal ^ ino AFE rij 

vxo VAE' oXrj uqk f} vno BAFSval raiq vxo ABF, 

AFB itST] iativ. iarX Sh xal t; vno ZAF ixtog rotJ 

15 ABF TptyraVov Sval ralg urco ABF, AFB ytoviaig 

i'ffjj" ietj aptt xttl fj V7tb BAF yavia tfj vno ZAP' 

OQ^ri apa ixaTepa' tj apa iv t^ BAF ^iiutvxXi^ 

yavia i; 11310 BAF dp-d'^ iativ. 

KaX ijtiX Tou ABr TQiyavov Svo ymviai aC vito 

SO ABF. BAF Svo OQ&mv iXatTovig eieiv, OQ&i} S\ ij 

vTt6 BAF, iXartav apa op^g ieriv 17 vno ABF 

ymvia' xal ieriv iv rp ABF (lii^ovi roi ^fiixvxXiov 

tfiJ^fiaTi. 

Kal iitiX iv xvxX^ tBrQaaXivQov ioti ro ABF^ 

1. loTio] (alt.) om. V. 2. Poat id add. aoxov m. rec. P7_ 
£] gapra hanc litt. eraB. T V; seq. in F: xul (m. 1) eU^tp^ 
IjiI Tfjs negiqiefcias (iu raa. m. 2) Svo TVxovva atiiista ta A, A 
(in mg. trausit m. 1); eadem omnia B inff. m. rec. ko^ — '&A\ 
in mg, tmDBit m.l F. %. AV,Aii,^r\ qi, seq. uestig.j<m, 1. 
i. 71 iwi BAr\ P; om. Theon (BFVp). 6. (H^om] -ow 
in ras. V; cort. es lalitov m. 2 B. 6. ABP] S in raa. V. 
7. fi vni A^r-\ om. p; mg. m. rec. B. 10. htC\ lutly P. 
11. AEE] P, F m. 1, V m, 1; EAE Bp, F m. 2, V m. 2. 



1 




ELEMENTORUM UBER m. 243 

Sit circulus ABFjd^ diametrus autem eius sit J3P, 
ceDtrum autem E^ et ducantur BAy AF, Ad^ ^T. 
dicO; angulum m BAT semicirculo positum LBAT 

jT rectum esse^ qui autem in seg-' 

mento ABT maiore^ quam est 
semicirculus, positus est; LABT 
minorem recto^ qui autem in 
segmento A^T minorC; quam 
est semicirculuS; positus est, 
^ LA^T maiorem recto esse. 

ducatur AEy et educatur BA ad Z. et quoniam 
BE — EA^ erit etiam L ABE = BAE [I, 5]. rursus 
quoniam TE == EA^ erit etiam L ATE =« TAE. ergo 
L BAT ^ ABT -{- ATB. uerum etiam angulus ex- 
terior trianguU ABT, L ZAT — ABT + ATB [I, 32]. 
itaque LBAT= ZAT rectus igitur est uterque 
[I, def. 10]. ergo angulus BAT in semicirculo BAT 
positus rectus est. 

et quoniam trianguli ^jBFduo anguli ABT, BAT 
duobus rectis minores sunt [I, 17j, et L BAT rectus 
est, L ABT minor est recto; et in segmento ABT 
maiore, quam est semicirculus, positus est. 

et quoniam in circulo quadrilaterum est ABT^, 



BJE}F; EBA Bp, e corr. FV. 12. FE] P; ^E F, V in 

ras. m. 2; EA Bp. EA] P; EF Bp, in rae. m. 2 FV. 

iatlv PB. xat'] om P. yoovia 17 FV (supra ycov/a in V 

ras. est). 13. FAE] in ras. m. 2 V. 15. ABF] (alt.) F 

in ras. m. 2 V. yoov^tgl m. 2 V. 16. Hcrji] (prius) m. 2 F. 
17. ASr P. 18. lctiv] PB, comp. p; iatt FV. 19. dvo] 
supra add. at m. 1 F. 20. ABT, BAr]ABr in spatio 6 

litt. m.2 F. iXdaaovig FV. 21. BAF] FFV; BAT ymvCa 
Bp. iXaaacav V. 

16* 



244 ETOIXEISN y', ■ 

tcov Sl iv Toig xvx}loi.s r£tQ«7ti,£V(f(ov aC aitevavriov 
ycjviai dvelv (Spdatg ("fffft tioiv [a! apa vno jiBF, 
A /1 r yavCai. dvelv 6(f&atg tCai ^iisiv}, xai tWtv 17 vxo 
ABF iXdrTfov opS-^g" ^oiJt^ apa r} vao Ajiir yavia 
5 ^Bi^tav oQ&ijs ietLV KaC ieriv iv ra A/JF iXarxovi 
rov tj(uxvx}.iov rif^fiari. 

Aiya, ort xal r] fttr tov ^eCtpvog Tfi^fiatog ytavia 
Tj XiQisxofiivi] vao [te] rijs ABF :ieQi<ptQsiag xal 
tijg AF sv&eCag (lei^av ierlv og&iig, fj Sh rov HaT- 

10 Tovog Tfi^fiaros ymvia ^ neQt,E%o^ivii vao [ze] t^s 
AAir] x£Qi(pEQeCag xal r^g AT evd^eiag ikaTTav ierlv 
OQ&ijg. xtti iariv avro&EV ^aveQov. iael yccQ ^ imo 
Tiav BA, AF ev&eiiov 6q&-^ ioriv, »j Kpa vno t^y 
ASr zeQi^eQeCag xal tijg AF ev&BCag XEQttj^oitivtj 

l'i fieC^aiv ietlv oQ&^g. ndkiv, insl ^ 11^:0 tiov AF, AZ 
Ev&eiwv op9'i; isriv, ij UQa vjto r^s FA BV&bCag xaX 
t^S A-J{r] neQig>SQBiag aeQiexOfiEvti ikarrav iaxlv 
OQ^rig. 

'Ev Kvxk^ aQU fj iiiv ev Tp ijfiixvxlCq) yrovia op#ij 

20 ietiv, i\ St iv T(B fieC^ovi r^^ftart iXdrrav OQ&^g, ^ 
di iv ra i}.dttovi [rftijftaii] ^eCtjav opd^s, xal ^rt 
ftEv Tov fti^ovog Tft^ftarog [yiDvCa] [lei^av [ietlv] oQi 






2. «r «pu — 3, slaCv'] mg. m. rec. P, 3. ywviai] 

Bp. lUlv] BF; tlel PVp. 4. loijcn] m. 2 P. ytovla] 

PF; om, BVp. 6, ope^s hziv] PF; oV^e ioii V; lanv 

dpfffe Bp, iaziv'] (alt,) om, V (aupra nal iv raa.). AJF] 
P, F, V (raB. eupra); om. Bp. hdxavi P. 7, Swl P, F 

m. 1; dii, 3x1 BVp, F m. 2 (euan-). 8. 11} P; om. BFVp. 
ABF] P; JHB P m. rec, BF, V m. 2, p m. 1; ABT cnm 
ras. 1 litt. inter ^ et B V m. 1 ; T add. p m. rec. 9. A T] 
r io ras. m. rec. B. fisi^io»'! fwi^J- in raa. m. rec. B. 10. 
te] P; om. BFVp, 11. AJF] F inaert, m. 1 F. ilatzaiv] 
in raa. m, rei;, B, 12, ii] 17 Titgttxofttvii yiovia V. 13. 

0^8"^] PFV (ia F ante 6if9ij inaer. ■aetjitiOjiivT} yaria mg, m. 



ELEB4ENT0RXJM LIBER m. 245 

et in quadrilateris in circulis positis oppositi anguli 
duobus rectis aequales sunt [prop. XXII], et angulus 
ABF minor est recto, reliquus angulus AdF maior 
est recto; et in. AAF segmento minore, quam est 
semicirculus, positus est. 

dico etiam, angulum maioris segmenti arcu ABF 
et recta AF comprehensum maiorem esse recto, mi- 
noris autem segmenti angulum arcu ^^F et recta 
AF comprehensum minorem esse recto. et hoc statim 
adparet. nam quoniam augulus rectis BA^ AF com- 
prehensus rectus est, angulus arcu ABF et recta AF 
comprehensus maior est recto. rursus quoniam an- 
gulus rectis AF, AZ comprehensus rectus est, an- 
gulus recta FA et arcu AAF comprehensus minor 
est recto. 

Ergo in circulo angulus in semicirculo positus 
rectus est^ qui autem in segmento maiore positus est, 
minor recto, qui autem in segmento minore positus 
est, maior recto, et praeterea angulus segmenti ma- 



1; idem mg. m. rec. P); nsQi6xo(isvri 09^17 ymvia Bp. 14. 

ABT] AHF P; AHB BF. V m. 2, p m. 1; F add. p m. rec, 
ABG cum ras. inter ^ et B V m. 1. AF] F in ras. m. 

rec. B. 15. fisi^mv] fisii- m ras. m. reo. B. 16. A F] TA 
V. BvQ^Bmv nsQisxofiivrj in ras. m, 2 V. 17. AJF] Ad 

P. iXdttmv] e corr. B m. rec, praeced. f m. 1; post ras. 

1 litt V. 20. iXuttmv ictlv BV. 21. tfi^fiati] om. PB 

FVp. fisiSayv iativ BVp. 22. ymvLa] om. P, m. 2 F. 
ictiv] om. P; m. 2 F. 



ETOIXEIftN y'. 



^ 



il Se tov iXiitTOvog r/iJjfHKios [yraviK] iKttTTO}v 
oaeg idet dsi^ai,, 

[IIoQiafia. 

'Ek tfij roilrow ^avtpov, ori iav [^] ftCa ycovia rpt- 

5 ymvov ratg Svelv fffij 7j, opfr^ ^Ortn ^ yatvia Sia 

TO xal r^v ixeivrjg ixTog Tatg avxaig tOtjv dvai ' iav 

(5i aC ifeifjg teai rotftv, OQ^aC ttat.v.'^ 

'Ettv xvxlov itpanTfjTttl vig ev^eta, a«6 9\ 

10 r^s Ky^s «^s TToi/ jewxAoi/ dtaj;^?; Ttff etyd^ef* 

rifivovOa rov xvxkov, ag iroier yravt'«s npog Tg' 

itpaTtxo^iv^, tStti eaovTttt tatg iv rofg ^vaAila. 

Toii xvxXov Tfiijfiaai yaviatg. 

Kvxlov y«p Toi; ABF^ i(ptt7JTsa9a rig evQ-ettt 
15 ^ EZ xara to 5 Srnistov, xal diio tov B atJiieiov 
tft^jj^*^ '*S Evftfiw f^s Tov ABF^ xvxXov zifivovea 
avtov 7) BA. i.iyin, oti ag noiet yQjviag ii Btd [lext 
Ttjg EZ iq)airTOfiivr}g, tetci laovrat raig iv rotg ivi 
Aeg tfirjfiaet ror xvxkov yaviaig, tovTietiv, ori ^ fii 
20 vxo ZB^ yavitt tarj iarl rfj iv rro BAA tfi^^fii 
ewteTttfiBvij ytavla, i\ S\ vTto EBz/ yavia [07^ ii 
zfj iv rp ^VB Tfi^^fittti evviaTafiivri yavia. 

"Hx&a yap ajto rov B rij EZ npog cip&ag ij B 



I 

1 



XXXII. BoetiuB p. 38B, li 



1. yiov£a] om. PBFVp. 3. Seq. alia demonatratio; 

appendix. 8, irrfpdifio — 7. eleiv] mg. m. l PFb; Bras. "V. 
4. oti] '/. F. ri] om. P, tpiyoitou ij (ila yiatia Bp, ^M 
Svo P. iazi B. ij fomia] Pb; om. BFii. 8. nai] e co 
F. ^Krds] Pb,Bm,rBO.; {rpe^ijs Fp, B m. 1. idv^Ph; S» 
FBp. 7. af] om, Pb. vtaviai Caai F. 8. tS' F; co 

m. 2. 9. Iqi- m.2 F, 10. ils zov kv-kIov] om.. F V. 



ELEMENTOBUM LIBER m. 247 

icris maior est recto minoris autem segmenti angulus 
mrior recto*; quod erat demonstrandum.^) 

XXXII. 

Si recta circulum contingit; et a puncto contactus 
in circulum producitur recta secans circulum^ anguli; 
quos haec cum contingenti ef&cit, aequales erunt an- 
gulis in altemis segmentis circuli positis. 

nam circulum ABFjd contingat recta EZ in puncto 
£, et a J3 puncto recta Bd circulum ABFjd secans 

in eum producatur. dico, angu- 
loSy quos B/l cum contingenti EZ 
efficiat, aequales fore angulis in 
alteruis segmentis circuli positis^ 
Ii. e. LZB^ aequalem esse angulo 
2 ui segmento BA^ constructo, et 
L EB/J angulo in segmento /JFB 
constructo aequalem. 

ducatur enim a ^ ad EZ perpendicularis jB^, et 




1) Gorollarium per se parum necessarium hic prorsus 
praue coUocatur, cum minime e propositione pendeat. si Eu- 
clides id adiicere uoluisset. post I, 32 ponere debuit. etiam 
collocatio uerborum onsg idsi. dsC^ai et ratio codicum inter- 
polatorem arguunt; omisit Campanus. post Theonem demum 
additum esse uidetur. 

Siaxd^l -a- in ras. V. 11. tijv itpaTttofiivTjv V; corr. m. 2. 
17. avto q>. 18. i(paiitofiivr}g'] -g postea add. F. 19. tov 
xvxilov t(M^iMiai. y. tfiT^fiaaiv P. oti] om. p. 20. ZBz/l 
z^BZ F; corr. m. 2. yawia] om. Bp. iativ P. iv t^] 
m raa. v m. 2. BAJ] PF, V e corr. m. 2; JAB Bp. 

21. yov^] seq. tj vno dAB, sed eras. V. EBd] d ia ras. 
V; dBE F, corr. m. 2. ymvia] PF, V in ras. m. 2; om. 

Bp. iativ P. 22. JTB] F e corr. m. 2 V. yoBVta] 

seq. Tjl vno JFB V (eras.), idem mg. m. 2 F. 



248 STOIXEiaN f'. 

■xai jWjjV^i» ^51^ ^^S 5^ ncQiipSQEias '^v%ov arjfittl 
To r, xal hcs^evx&toaav at AA, ^F, FB. 

Kttl in£l Kvxkov xov ABF^ etpaxrsTai tis ev&s\ 
^ EZ xatci TO B, xal aab tijs apiig ijxrai TfJ i^ 
6 anroiiivrj tCqo^ oQ&ag 17 BA, fxl t^s -B-^ *^&^ 
xivTQOv ierl ToiJ ABFjJ xvxkov. t/ BA «pa (^«ffte^l 
tQog iaxi roii ABT^ xvkXov t; Kpa •vno A/IB yta- 
v(a iv tjfuxvxkia ovaa opftTj i6ziv. Aotffat «pa af 
ujro BAd, ABd fua op^^ fffat f^fl/v. effrl 8h xal 
10 ^ ijjro ABZ opd'ij' 17 apa ywo ABZ fOij ^«t1 
ii«6 BA^, ABA. xoivi] aqpijp^ffffo) ^ ureo AB^\ 
i.oiJcij aQU ij vno jdBZ yavia fO»j ictl Ttj iv rp 
aAAa| TfnJfiaTt rov xvxkov yiavia tjJ uko BA^. 
i^d iv xvxka TiTQaicXtvQov iOTi t6 ABPA^ ut a: 
16 tvavrtov ctvTov yaviai SvoXv oQ&atq taat sisiv. elt 
di xal aC v%o ABZ, ABE Svalv opffats f»at 
11310 JBZ, JBE Tttbg vno BAJ, BTJ iGai, Blalvy 
wv i} v7to BAA T^ vTto ABZ iSiijpti laij' kou 
UQu ii vno ABE tij iv Tp ivaki.efi tow xvxlov tfii 
20 fiOTi T<p AFB TJJ imo JTB yavia ietlv Hai}. 
■ 'Eav a.Qtt xvxkov iipamritai rtq ev&Bta, aitis 

H T^s aipiis dq tav xvxXov Siai&ii tiq ev&eta rifivowta 

^L Tov xvxkov, ag noiEt ytoviag xqos t^ iyajwofiivpj 

^^k taai ieovtai tats iv tots ivttkkal tdv xvxkov tfttj^atn 
^^1 S5 ymviais' oxBQ iSst Stt^t 



I 



1. Et/] in raa. m. 1 P[ inter B et J inaert. T m. 2 F, 
2. J r, rS] litt. rrB in ras. m. 2 p. 4. x«! ojio] cwo di 
P. t^e] P; T^s xaiM 16 B Theon (BFVp), 6. BA] (bii) 
AB F. 6. imiV P. 6. ii BA — 7. waou] om. Bp. 7 
htiv P, ut Un. 0, 10. 12. 14, _ fi agaii V. 6. Iflni'] P^ 
comp. p; iaii BF, 9, fiic op^^] mg. P, 14. ul] ual 4 

PV, 16. yaviai} poBt hoc uooabulum in FV n " " 




ELEMENTORUM LIBER m. 249 

in arcu Bj4 sumatur quodlibet punctuin Fy et ducantur 
4^y dFy FB. et quoniam circulum ABFjd contingit 
*ecta EZ in B, et a puncto contactus ad contin* 
;entem perpendicularis ducta est BA, m BA centrum 
jrit circuli ABF^ [prop. XIX]. itaque BA diametrus 
5st circuli ABF^. quare L ^^B, qui in semicirculo 
30situs est^ rectus est [prop. XXXI]. ergo reliqui 

BAJ + AB^ 
mi recto aequales sunt [I, 32]. uerum etiam L^BZ 
ectus est. itaque L ABZ = BA^ -\- AB^, sub- 
rahatur^ qui communis est; L^Bj4, itaque 

LdBZ = BAJ, 
[ui in altemo segmento circuli positus est. et quo- 
liam quadrilaterum in circulo positum est ABFdy. 
•ppositi anguli eius duobus rectis aequales sunt 
prop. XXII]. sed etiam /.^BZ + ^^BjB duobus 
ectis sunt aequales [I^ 13]. itaque 

^BZ + JBE = BAJ + Br^, 
[uorum LBAd = dBZy ut demonstratum est. ita- 
ue L ^BE = jdFBy qui in alterno segmento circuli 
dFB positus est. 

Ergo si recta circulum contingit; et a puncto con- 
actus in circulum producitur recta secans circulum, 
nguli, quos haec cum contingenti efficii, aequales 
ruut angulis in altemis segmentis circuli positis; 
uod erat demonstrandum. 



t vno BAJ, JTB. 16. stal Si — 16. taat] P (slaiv)] om. 
heon (BFVp). 17. JBZ]\iit JB e corr. m. 1 F. In 

seq. mg. m. 1 : at slai, dvalv og^atg taai dia xo sv^stav xiiv 
\B In Bv^Biav {-av non liquet) njv EZ mg itvxs saxdvai, 
I. TOis] insert. m. 2 F. 



250 STODtEKiK y'. 



ly. 

xvxlov Sbx6ii£vov yavCav fffiji' r^ SoQ^sCt 
yavCa iv&vypttfifie}. 
6 "EeTm ^ So&stea iv&iia 57 AB, i{ Si So^staa 
vCa 6v&vyQa[i[ios r] ffpog ra F" dff Srj inl trjg 
&tC6i}s tv&iCag rijs AB yQaipai Tfiijfia xvxlov Sej 
(igvov yatvCav lerjv T/j wpoff ra F. 

'H Sij ffpoff Tm r [ytavCa] ijroi o^tld iQxiv rj dpfi^ 

10 1? a[t^i.tia' iffTw npoTspov o^iia, xal tog ^kI t^g n^fo- 
TTjs nKraypagj^g euvfOirKTro jrpog r^ AB £v9sCa xal 
TM y4 ffijfif^G) T^ ffpos Tp r* yavCa tatj ^ vno BAjd 
o^sta a^a ierl xal ij vito BA^t. ijx^'^ ^jj ^A jrpog 
opfl-aij ij ^jE, xal TstfiijaQa ^ ^B d^X"" 5""^^ t6 Z, x{;J 

15 jjjjfl^io aito Tot» Z er}[i,{Cov tjj AB «pog opOtcg ij Z. 
xul ineUv%Qa ij HB. 

Xal iTtsi i'6r] i6tlv ^ ^Z tij ZB, xoivi] Si ^ Z 
tfwo d^ Kf _rfZ, ZH Svo tatg BZ, ZH taai Eieli 
xttl yavCa ij vjco AZH [yavCa] tj) vitd BZH fffi 

20 §d6is «pa 17 y4if ^affEt rj; BH fffij ^ffri'!'. 6 apa xA*^ 
rpoj (lev rra if Staffr^^aTi S^ ra HA xvxXoq ygatpo- 
yiEvog ^'isi Kttl Sia toij B. ytypaqid-a xal IffTo o 
ABE, xal imttvx&a ij EB. intl ovv a% axQus t^g 
AE Sitt[ieTpov ano roii A r^ AE ^igog op&as iotiv 

XXXin. [Euclid.] opt. 47 (Stndien p. 122). Simpliciua im 
phjB. fol. 14. Boetius p. 388,20—21? 



I 

xal 

'^- 

I 



L 



1, It' ¥. 6. Ti] (primum) om. p. 8. r^] t^ PF. T] 
i r vio»'^a Theon (BPVp). 9. *n] Bcripsi; «i P; npa m.3 
V; yrip 'Bp, F m. 1. yWoT P; om. BFVp; in F 

Id. m. rec. ^] flupra scr. m. 2 V. 10. ngotcgov] jtpw- 

w V. xal soe] P, F (ho:(' del. m. 2); as Bp, e corr. V. 



ELEBIENTOBUM LIBEB m. 



251 



xxxm. 

In data recta segmentum circuli construere, quod 
mgulum capiat aequalem dato angulo rectilineo. 

Sit data recta ^iBy et datus angulus rectilineus is, 
jui ad F positus esi oportet igitur in data recta 
4B segmentum circuli construere^ quod angulum ca- 
piat aequalem angulo ad F posito. 

angulus igitur ad F positus aut acutus est aut 
*ectus aut obtusus. sit prius acutus, et, ut in prima 

figura^ did AB rectam et punctum A 
construatur angulus aequalis angulo 
ad r posito LBAJ [I, 23]. itaque 
LBAd acutus esi ducatur di,A dA 
perpendicularis AE^ ei AB in duas 
partes aequales secetur in Z^ et a Z 
puncto 2i,AAB perpendicularis 'ducatur 
ZHy et ducatur HB. 
et quoniam AZ = ZB, et communis est ZH^ duae 
ectae AZ^ ZH duabus BZ^ ZH aequales sunt; et 
^AZH-^^-BZH itaque AH-=-BH [I, 4]. quare 
irculus centro H radio autem HA descriptus etiam 
)er B ueniet. describatur et sit ABE, et ducatur EB. 
am quoniam ab A termino diametri AE B.d AE per- 




1. tiUTacTQOtp^s 7* icff^ avvsocdtm Bpqp; tial om. P, m. 2 

^ 12. A arifis^o}] ^r^os ttvr^ arjfjLsioi t£ A V. 13. kaxCv 

*F. xal ^x^cD bp. JA] AJ iYp. Dein add. dnh 

ov A arinBCov Bp, P m. rec. 14. AE] E in ras. Y. xal 
STttiiad^a rj AB]mg. m. 2 F. 18. $vo] (alt.) dvai Vp. 
IZ] ZB Bp, F V m 2. dai Vp. 19. yoQvta] P; om. BF Vp. 
BZH]Fi HZB Bp, V (sed H et B in ras.); ZB supra scr. 
r m. 1 F. tarj iattY. 20. BH] HB F. 23. EB] BE P. 



252 STOIXEIQN y', ■ 

^ ji^, 71 AA a'p« ig>ttXT£T:at Tov ABE xvxlov' ixft 
ovv xvxXov tou ABE i^ajizszai rig tvd^eia ij A^, xal 
«Ko T^s xaza ro A atp^g siq rhv ABE xvxXov StijxtaC 
ns ev^sta tj AB, ij ap« viro ^AB yfovia tGri iorl 
5 T^ iv rp ^ftiAAal roti xvxkov Tfiiljtati ytovia tJ} vao 
AEB. alX' 1] vjto -dAB trj ffpo^ ta F i&ttv CtfTj' 
xal f] rcpog ta r «pR ymvia tet} iarl r^ vxh AEB. 
'Enl t^s tfoSstoJjg «pa Ewdffos t^s AB tfi^fia xw- 
x/ou yiyQantai ro AEB dEXOftevov yaviav rijv 'ftxo 

'AXXa 6ij OQ%ii larai ij nQog ta P' xal Siov 
Xiv ieta htX r^s AB yQu^w Tfi^fia xvxXov Sixofievov 
yasviav Hariv ty tcqos tro P op&^ \ytovi^}. CuvEiJiarci 
[aaXiv] tij ngog tS F op9^ yasvia l^eij ij vno BAjd, 

15 atg i%Ei inl Tijg SBvtBQag xaTayQuip^g, xal tst(n^6&a 
1} AB S£%a xata to Z, xal xivtQGt tp Z, SiaaTrj- 
ftart d% oJtoTEpra tmv ZA, ZB, xvxXog yEyQaip&m o 
AEB. 

'Etpamstai. aqa f) A^ sv&tta toij ABE xiixXi 

20 dta ro 6Q9r]v slvai tijv jrpoff rp A yaviav. xaX 
ietlv i] vno BA^ ytavia r^ iv rra AEB tjf^fiatt' 
OQ&il yap xal «utij iv ii[itxvxXia ovea. aXXa xal 17 
vno BAj4 tfi Jtpog rw r terj iativ. xal ij iv rp 
AEB «pce i'ari iarl Tf} srpos tra F. 

1. ^EB] om. Bp; supra eat raa. in V. UtX orv] PPTP 
(yp. ;(ot ^itii' F mg,), xol ^icsi; Bp. 2. sou .^BE mfnloft ^ 

Bp. .<BE} .iEB e corr. V. 4. JoiC» PB. 5. h r£\ 

om. P, 6. aUa P. iJ.<B] litt. .i.,1 in raa. m. 1 P, dein arfd. 
r^i ijio ..iEB, del. m. 1. 7. inlv P. 8. IwJ] -t e cotr. 

m, 2 V. ^B] .i eras. p; t^^^ %v%Xov F. 9. E,<<B F. 
10. r^l (alt.) om. F. 11. ?oroj naXiv P. 13. ymvial F; 
Om. B^Vp. 14. jidlif] F; om. P; yap ndXtv BVp. ' 16. 

fiiv r<5 V. 19. .iBE] corr. ei ^BF m. 1 P. 20. yojTia»] 



f 01» * 

arci 
^^, 

frf 
ni 

ttt' 

?« 



ELEMENTOBUM UBEB ni. 253 

pendicolaris dacta est AJ^ recta A^ circulum ^jB £ 
contingit [prop.Xyi srdp.]. iam quoniam circulum ABE 
contingit recta A^^ et ab A puncto contactus in circu- 
lom ABE producta est recta AB^ erit L ^AB = AEB, 
qniin altemo segmento circuli positus est [prop. XXXII]. 
aerom L AAB angulo ad F posito aequalis est ita- 
qne angulus ad F positus angulo AEB aequalis est 
ergo in data recta AB segmentum circuli AEB de- 
scriptum est^ quod angulum capiat AEB angulo dato^ 
qui ad F positus est, aequalem. 

iam uero angpilus ad F positus rectus sit. et rursus 

propositum sit, ut in recta AB segmentum circuli 

describatur, quod capiat angulum recto augulo ad F 

I posito aequalem. construatur rursus 

Jn :^ angulus BAA recto angulo ad F 

^^ posito aequalis, ut in secunda figura 

factum est, et ^^^ in Z in duas partes 
aequales secetur^ et centro Z radio 
autem alterutra rectarum ZA^ZB cir- 
culus describatur AEB. itaque recta 
A A cironlum ABE contingit^ quia angulus a,dA positus 
rectus est [prop. XVI xoq.]. et LBAA angulo in seg- 
mento AEB posito aequalis est; nam hic et ipse rectus 
est, quia in semicirculo positus est [prop. XXXI]. uerum 
L BAA etiam angulo ad F posito aequalis est. ergo 
etiam angulus in segmento AEB positus aequalis est an- 

m. 2 V. foiy] PF; om. BVp. 21. tfii^fjLaxt tari BVp; supra 
TfftiJfMXTt in F duae litt. eras. (y»?). 22. iv] m. rec. P. 

%€c£] PF; om. BVp. 23. iativ Haii BVp. Ka^— 24. t« 

rj om. Bp; supra est ras. in V. 24. AEB] m ras. m. 2 V. 
Dein add. xfkriyLazi P m. rec. tan iaxi] P {iatCv)\ om. V; ras. 
6 litt. F. r] P, F m. 1 ; i'an iatCv add. F m. 2; T iaxiv ian V. 





264 STOIXEIiiN y. 

ViyifaTirai apa itakiv inl t^g j4B tfiijfia xifxXov 
ti} AEB Sb%6iuvov yavCav l0^v t^ jrpog ta F. 

AXXa drj rj Jtpoj tra T aii^i^eta ^otco' xal Ow- 

eazaTiD avzij lari ^ipog rf) AB bv&eCu xal x^ A uji- 

5 fiB^p 57 tfiro BA^, rog ^^(fi ^rol r^s tQCzris xaTayQtctprjs, 

xal TJ3 A^ jrpog opS^ag ^j;*''' ^ ^^j xaX rfiTftijeffi» 

%dhv 7} AB Si%a xata to Z, xcc^ tfjAB irpog dp9i 

Kai dntl naf.lv tarj iatlv r^ AZ t^ ZB, xal xoii 

10 i} ZH, Svo di] at AZ, ZH fivo tats BZ, ZH ftfoi 

slaCv xal yrovCa i) vnh AZH yavCa rij vno BZH 

larj- ^datg apa rj AH ^datt t^ BH tarj iaTiV 6 apa 

xivTpci jilv ta H Sia6Trjftati. Si ta HA xvxXos yqu- 

(pofisvos ^|e( xal Sia rot' B. fpjjEffOto ms o AEB, 

15 xttl insl tij AE SiafiETQm ait axffag jtpog op^«s iotiv 

rj AA, f] A/S UQK iqidTitETai roO AEB xvxi.Ov. xal 

a^ t^g xttTK t6 A inagiijg Sirjxtai 17 AB' tj a^a vab 

BAA yavCa i'ar] iazl ty iv ttp ivaXXd^ rou xvxXov 

t(iij(iati. rra A0B avviataftivri yavCa. aXX' ^ vxo 

20 BA/i ymvCa tij ngog rta r* Car) iaTiv. xal »; iv TJ» 

A®B apa Tfi-^fiart ymvia Carf iatl f^ fflpog Tp V. 

'Exl trjs aga do&iCaris tv&ttag T^g AB yiyQanrat 
Tfi^/io: xvxXov tb A&B Ssxof^Bvov yovCav Carjv 
jrpog Tco r- ostBQ iSti noi^aai. 






2. ABEV. r oe*n V. P m. reo. 4. foi;] 
ji] in Bvrji m. 2 snpra Bcr, P. 9. ZB] in tas. r. Kat 

xoivTJ] xotv.; ^E PV. 10, ZH] (alt,) H in ras. m. 1 B. 

Wo] PB, ivSi P m. 1; Svai Vp. U, t^oi.' V)p. 18. Post 

r<T7j ftdd, ^oi^ V, F m. a. 13. IIA] corr. ei ^ m. rec. P. 
16. |jre^] corr. ei In^ m. 2 F. ^ot.»] P; cfr. p. 260. 21; 

iJKiai Theon (BFVp). 16. AEB] MttEB m rae. F. 17. ;}] 
(ptins) in ms. m. 2 V. 18, /or.V P. 19. ASB] Utt. 68 



I 



ELEIIENTOBUM LIBER m. 



265 




^lo ad r posito. ergo rursas in ^iB segmentum circuli 
escriptum est AEB, quod angulum capiat aequalem 
ngulo ad F posito. . 

iam uero angulus ad F positus 

jps^ obtusus sit, et ad rectam jiB et 

punctum A ei aequalis construatur 
LBAdy ut in tertia figura factum 
est, et ad Ad perpendicularis du- 
catur AEy et rursus AB in Z in 
duas partes aequales secetur, et ad 
AB perpendicularis ducatur ZH^ 
t ducatur HB. et quoniam rursus AZ = ZBy 
t ZH communis est, duae rectae AZ^ ZH duabus 
iZj ZH aequales sunt; et L AZH^ BZH. itaque 
iH = BH [I, 4]. itaque circulus centro H et radio 
lA descriptus etiam per B ueniei cadat ut AEB, 
t quoniam ad diametrum AE in termino perpendi- 
ularis ducta est A^, recta A^ circulum AEB con- 
ingit [prop. XVI tcoq,], et ab A puncto contactus 
iroducta est AB, itaque L BAzl angulo in altemo 
egmento circuli, A&B, constructo aequalis est [prop. 
lXXTT]. sed LBAd angulo ad F posito aequalis est. 
[uare etiam augulus in A&B segmento positus angulo 
,d r posito aequalis est. 

Ergo in data recta AB segmentum circuli con- 
tructum eBiA®By quod angulum angulo adFposito 
.equalem capiat; quod oportebat fieri. 



1 ras. m. 2 V. evvBisxaikiviii PF. alXd P. 20. kaxl V. 
21. ymvia] om. V. ietCv P. 22. aqa io^siafig] PF; 

o&BCarjs «9« BVp. AB] in ras. FV. 23. S s xo iisv ovj corr, 
X ixofisvov m. 1 P. 



id'. 

'A^o Tot^ So&ivTos xvKkov Tftijiia dipEXeB 
i}4j;of**»'oi' yaviav larjv Tfj SotfdCtS'^ yavi^ i 
^vypafifiG}. 
5 "ESTfD 6 So&dg xvxKog 6 JBF, ij Sl So&etaa 5 
via sv&vyQa[i[ios ^ ^rpos ™ -^ ' ^*f 'J') aitb tov j4 Ji 
xvxlov Tft^fia aqtEkitv Stji^ofisvov ymviav ta^v zf) SS 
QeCerj yavia sv&vyQafifie) zfj irpog ti5 ^. 
"ifx^ia totJ ABF iffanTOfiivri ij EZ '. 
10 ejjfiBtov, xal evviaTttTC3 jrpog t^ ZB svd^sia xal xm 
stQos KWTJj arjfitio} rm B t^ repog Tp ^ ymvia fojj ^ 
VJTO ZBr. J 

'Eail ovv xvxioi' roij ABF itpdntiTat rts £v#nH 
^ £Z, xal ana Trjg xara xo B inacp^g dtijxTai ^ BfJ 
15 ^ liroo ZBF apa yavia tari iatl tj; ^ji za BAF ivaiXtt^ 
Tfi^fiKTt evvieTafisv'^ yovia. aKX i] vxo Z BT Tjj 
nQog ta J iGtiv fffjj" xal 7} iv ta BAV aga tfi^ij^ 
[ittTi taij iatl tfj Jtpog rp .^ [yaviai. 

'Ano Tow do^&^i^os «pK xvxkov roi; ABT tfi^m 

20 a^jJpijTai ro BAT SexofLEVov ymvCav taijv rfj So&sU 

yatvCtt EV&vyffdftfLcy x^ «(fog rra ^' omg iSei stot^Oi 



Affj 



'Eat' ^v xvxX^ ffvo fud^erai tifivaaiv di,i« 
j, To uJTo TtDi' T^s liiag TiiijftdTmv 3csQie%4 



1. Is' F. 6. SiC cSij — 7. diifslsi*) om._F; add, m,] 

mg. 7. ywv^ ip. rn io^iiejj ytoyia ev&vycdfifito] P; o 
Theon (BFVp). 8. JJ J ymWa Bp,' F m. 2, V m. 2. 
ABP xvkIov V, sed xiixlou punciis notat. ^] (vd^cro: ^ 1 
F m. rec. B] corr. es r ra. 2 fc'. 10. ZB] BZ P. 1 
tmj (alt.) rn p; oorr. ra. 2. 13. ABF xKta to B T, P n 

leo. nsl m, 2 F. 16. ywWo] ora. Bp. fiH] Joii'] 00 




ELEMENTORUM LIBEB lU. 257 

XXXIV. 

A dato circulo segmentum aaferre^ quod angulum 
capiat dato angulo rectilineo aequalem. 

Sit datus circulus jiBFy 

et datus angulus rectilineus 

is^ qui ad ^ positus est. 

oportet igitur a circulo 

y^^Fsegmentum circuli au- 

ferre, quod capiat angulum 

aequalem dato angulo recti- 

lineO; qui ad ^ positus est. 

ducatur EZ circulum ABF contingens in puncto 

By et ad rectam ZjB et punctum eius B angulo ad ^ 

posito aequalis construatur ZBF [I^ 23]. 

iam quoniam circulum ABF contingit recta EZ, 
et a puncto contactus B producta est £F, LZBF 
aequalis est angulo in BAF altemo segmento con- 
structo [prop. XXXII]. uerum L ZBF angulo ad ^/ 
posito aequalis est. quare etiam angulus in segmento 
BAF positus aequalis est angulo ad ^ posito. 

Ergo a dato circulo ABF segmentum ablatum 
est BAFy quod capiat angulum aequalem dato angulo 
rectilineo, qui ad /t positus est; quod oportebat fieri. 

XXXV. 

8i in circulo duae rectae inter se secant, rectan- 

V. BAF^ BA Q corr. m. 2 V; .4Br F. 16. avviatccaivjj 
F. yoivCa Ccrj iaxCv V. t^] ymvia tari iatt t^ V. 17. iativ 
tarj] om. V. tfn^fiati] P; tfii^fiati ymvla Theon (BFVp). 

18. iazlv P. yoavia] P; om. BFVp. 19. tcw] (alt.) om. 

F. Tfi^/iia Ti V et corr. ex xfirifiati F. 22. X € ] euan. F. 

Eaolides, edd. Heiberg et Menge. 17 



258 ETOIXEiaN /. ■ 

fievov oq^oymviov tuov ietl rp vao tcov tijs 
itigag rfuj/taTov ffSQiexoiiivm oif&oymvip. 

'Ev yct^ xvxkat rra ABF^ Sva iv^&tai at AV-, 

B^ tBp.vitoaav alKrjlag xata ro E eTjiiiiov kiyta, 

h ori t6 vTto tmv AE, EF a£gt£x6[iii'ov OQd^oymviov 

i'6ov ietl t^ viio tav jdE, EB xepiBj^Ofiivo} 6if9v- 

ywvlip. 

El }xtv ovv al AF, B^ Sta tov xivzQov eIo\ 
caetB ro E xivtQov ^lvai Tov ABFA xvxXov, (pavi 

10 pdv, ori teav ovSmv tmv AE, Er, JE, EB xal 
vnb tav AE, Er iit(QiB%6nEvov OQ&oyaviov leov it 
rp vao Trov AE, EB neQisxofiiv^ dp&oyavCw. 

Mri etsraffav dij aC AP, ^B Sia tow xivtQov, xaX 
dX-^fp&G) t6 xivrQov tov ABF^, xal Setia t6 Z, »al 

15 Kffo Tov Z inl tag AF, ^B Bv%e(as xa&trot ^x^^ojffar 
at ZH, Z@, xal ixet^vx&coeav ai ZB, ZT, ZE. 

Kal insl avQ-fia rtg dia rou xivtpov 7} HZ bv- 
&tittv ttva (lij Sia xov xivtQov rijv AP irpo? o^^ag 
rdiivsi, xai SCxa avri^v ri^vet' fffij aga tj AH ty HT. 

20 inel ovv tv&Bia f} AF tirfttitai sig (tiv l'9a xaza to 
H, slg S\ aviSa xaTn t6 E, t6 «pc vtco rav AE, EF 
nBQiBxoiLEvav op^oytovtov ftsta tov ano tijg EH te- 
tQttymvov fffov iffrl ta a:r6 T^g HF- [xon'6i'] xqoe- 
xsia&ca to kje6 t^s HZ' to «pa vno tmv AE, EF 

26 (ista rav ano twv HE, HZ teov iotl rotg ajib rav 
PH, HZ. aUa totg fisv anb rav EH, HZ feov 
ierl tb Aab t^s ZE, toig 6h dnb tav FH, HZ teov 



*o- 

i 



3. yoe] yae to BFVp. AT.BJ] litt.r.B inraB.m.a Vj 

r,BJ in ras. m, 1 B; Ar,iJB F. 6. t(ov] om. P. 8. B^] 

^B F. eteiv] (oaiv V. 10. EF] in «w. m. 2 V. 13.ji^ 

Imatoav Sri] P,F (mg.m. 2: yp. ^orcoca» *^); fDKBouv S^ BVp, 

AT, JB] litt. r, JB iQ ras. m. 8 V. S,a] PF, V m. 1, p 



^^^ 




ELEBIENTORXJM UBER m. 259 

golam comprehensum partibus alterius aequale est 
rectangolo comprehenso partibus alterius. 

nam in circulo ABF^ duae rectae AF^ Bd inter 
se secent in E puncto. dicO; esse 

AE X £r -= ^E X EB. 

iam si AF, Bd per centrum duc- 
tae sunt^ ita ut E centrum sit circuli 

Jli[ ^>^^ ]j ABFAy manifestum est^ esse 

AE X Er — AE X EB, 

cum aequales sint AE, EF, ^E, EB. 

ne sint igitur AFy AB per centrum ductae. et 

sumatur centrum circuli ABFzly et sit Z, et a Z ad 

rectas AF, AB perpendiculares ducantur ZH^ Z^ et 

iucantur ZB, Zr, ZE. et quoniam recta per cen- 

trum ducta HZ aliam rectam AF 
non per centrum ductam ad rectos 
angulos secat, eadem eam in duas 
partes aequales secat [prop. IIIJ. 
itaque AH= HF. iam quoniam 
recta AF in partes aequales diuisa 
est in H, in inaequalss autem in 
B, erit AExEr+ HE^ = HJT* [11,5], commune 
idiiciatur HZ^. itaque 

AExEr+ HE^ + HZ^ = FH^ + HZ\ 
aerum ZE^ = EW + HZ^ et 

n. 1; pkh did B, V m. 2, p m. 2. xa^] mg. m. 2 F. 14. 

4Brz/]litt. rz/inras. m.2 V. Dein add. nvxlov P m. rec, P 
aostea insert., V m. 2. 17. HZ] ZH F. 18. fti}] poetea 

nsert F. 19. tsfivsi] (alt.) PFV; tsfist Bp (F m. 2). 22. 
HE V m. 1, corr. m. 2. 23. HF tstQaymvm V. xoivov] 

)m. P, post nqoansCa^oa add. m. rec. 25. HE, ffZ] alt. H 
\ corr. m. 2 V; Zff, HE P (ZH corr. ex ZE m. rec). Cea 
?. ictCv PB. 

17* 




260 ETOIKEiaN y'. W 

iezl To dao r^s ZV- z6 tpor vm rav AE, EF (i^a 
tov ttjro r^e ZE fffov iarl tm aTto r^g zr. t6r] H 
il Zr tij ZB' xb ttQa vno zav AE, BF [itta tou 
Kjro r^g EZ teov iozl tm azo t^g ZB. Sta ta 
6 avttt 6i} xal to vito zmv ^E. EB [iBra tov dno xijg 
ZE [eov iazl tra ttJTo rijg ZB. ideix^rj 8h xal ro 
imb rmv AE, EF fista tov aTib f^g ZE tOov t^ 
ano T^s ZB " ro apK vno tav AE, EF fieta rov dxo 
T^S ZE taov detl ta wto rcof ^E, EB |U£rK roti 

10 dnb t^s ZE. xoivbv d^pjjg^^e&o} tb ditb t^s ZE' 
Aotreov ap« ro vxb tiav AE, EF it£ffisx6[iEvov op- 
^■oyinviov tSov iozl ta VTtb taiv dE, EB ^CEpiE^o- 
[liva oQ&oyavia. 

'Edv a^a Iv xvxla sv^stai dvo ti(tvmeiv di.i.-qkagy 

is ro V7tb rmv x^g [iiag r(iT][idtoiv Ttepisxof-svov op#o- 
ymviov tOov ierl rra ujro r<av r^s ET^pag rft7][ittte}v 
xsQtsxofidvN 6(f&oya)vio}' oxeg edei iHst^ai. 

Ag". 

'Eav xvxlov Xti<p9^ ri eijfisCov dxtog, xal' 

20 aa' avToi Ttpbg tbv xvxlov ■jtQoaitijct&ei Svo 

sv&etai, xttl ^ [lev avrmv rdfivjj tbv xvxXov, 

i] dl iipdarijrtti., BSzai. rb vxb okris r^g refivov- 

etjs xal z^g ixtbg daoXafifiavofidvrjg (isra^v 

Tou te BTifieiov xal tijg Kupr^s xepnpsffeittg 

26 teov tw dnb T^g iipaxtofidvtjg tszQayav 

KvxXov yuQ tov ABV EtKtitp^a ti e^fietov dxTi 
tb A, xal dxh tov /4 npos roi' ASF xvxXov 

6, kBelx&ii 84] coffM Pi mg, m. rec; yo. iSUx»ri Se. 
iStix^ — 6' ZB] om._p. 11. ne^iPtoiifrov dpd'oyio»ioi'] 
m. S V. 13, T^] zo qs, 15, vno z^s fiiuf Tmv P. 



i 



ELEBIENTOBUM LIBER in. 861 

Zr^ — FH* + HZ* [I, 47]. 
itaque ABxEr + ZE^ ^ ZF^. sed ZF^ZB. 
itaque AE X EF + EZ^ = ZB\ eadem de causa') 
erit AE X EB -\- ZE* = ZB^. sed demonstratom est 
etiam AExEr+ ZE* = ZB\ itaque 

AE X Er + ZE* = AE X EB + Z fi«. 
Bubtrahatury quod commune est^ ZE\ itaque 

AE X £r = AE X EB. 
Ergo si in drculo duae rectae inter se secant, 
rectangulum comprehensum partibus alterius aequalc 
est rectangulo comprehenso partibus alterius; quod 
erat demonstrandum. 

XXXVI. 

Si eztra circulum punctum sumitur, et ab eo ad 
circulum adddunt duae rectae, et altera harum circu- 
lum secat^ altera contingit, rectangulum comprehensum 
tota recta secanti et parte eius extrinsecus inter punc- 
tum et partem ambitus conuexam abscisa aequale erit 
quadrato contingentis. 

Nam extra circulum ABF sumatur punctum A, 
et a ^/ ad circulum ^^jBFadcidant duae rectae ^FAy 

1) BB = SJ (prop. III). BExEJ + ES^ =- B©* (11,6). 
BExEJ + ES^ + Ze* ^ B©«+Z©* = JBZ* 
= BJExE^ + ZE« (1,47). 

tfjkJlfuitmv] tmv tfirjfidtaiv p. 17. onsQ iSsL dfigat] otisq tp. 

18. Irf F; corr. m. 2. 20. «Qocnintmeiv P. 22. I^ffrttt] 
om. FV. trjg oiijg t% ^, F m. 2. 24. nsQifpsQBiag] PBFp; 
add. nsQiBxofisvov OQ^oyonviov V, F mg. m. 1. 25. taov 

ict£ FV. 



fihv 
yto- 

zh 



262 STOIXEIiiN y\ 

xaczhaSav Svo sv&Elat aC jdr[A)„ ^B' xal 57 ftiv 
/ITA Tiiivhm xhv ABF xvxlov, ■}] Sl BJ ifpaXTi69ai 
liyto, 0T4 ro vito rmv AiJ, ^ F asQi£%6(iEvov OQitoym- 
viov teov ioil T(5 dao tijg AB TET^aycavei. 

(i 'H apa l^yrA ^TOi dia roij xivrQov iorlv ^ 
ffTro Tt^oTt^ov 6ia tou xivrgov, xal iOTa ro Z xiv" 
TQOV toii ABT xvxXov, xal Bns^tv%9et ij ZB" OQ^ 
aga ierlv 1/ vno ZB^. jtat eml sv&Eia tj AF Si%a 
rityLTirai xoitk to Z, itQoaxtirai Sh avz^ ij r^. 

Kpo; V7t6 zmv A^, AT niza xov uTto i^s Zf taov- 
iazX Tp oino zrjs ZA. /"Utj di ^ ZjT t^ ZB* ro £( 
■uKO TWi' ^z/, ^J" ft£Ta rou ttiro T^g ZB lOov itftl 
za dzo r^s ZjJ. tki 5i awo t^s ZA i'6a iatl ri 
arao rav ZB, BiJ' zb UQa vito tmv A/i, JlT (wra 

& rou aTto rijff ZB tGov iGtX rotg eiro tiov ZB, BA. 
xotvof a^^fj^ija^G) To ajio r^g ZS" Aomov apa to 
vao Ttov AA, AT tOov iezl rto dao Ttjg AB itpait- 
zonivrjs- 

dkXd Sij Tj ATA (lij iara Sia zov xivZQOv rov^ 

ABT xvxXov, xtti £li.rj<p9a ro xivzgov t6 E, xal dxh 
TOK E ijil rriv AT xd^czos ^X^"' ^ ^^' ""^ *''^*' 
^svx&aeav al EB, ET, EA' opS-^ aga iffriv ^ ureo 
EBA. xal iael sv&std rig Sid tov xivTQov tj EZ 
6v9'6tav riva /tij Sia rou xivT^ov ttjI' AT n^os OQ- 
26 &ag zifivsi, xal SCy^a avzriv tifivtf rj AZ apa tfj 
ZT iaziv tat]. xal insl sv&sta ij AT rfrftijrat Sixt 



1. jr^] Jr P, P (postea insert. ^). 2. JB B. 3. J 
in raa. p; J id im, m. E V, inBert. m,*2 B, m. xei 
r ¥; corr. m.2; Fd in ras.p. 6. &Qa] om.BFVp. JT* 
FA P, Jjr F, eed corr, 8. AF] r e corr. m. 2 V". 
^*;] J inraa. m.2 V. JT] supra m. 2 F; F P, corr. m. 
Toi UTii xiii] to VKO F; corr. m. 2. 11. Z J] ZA F? 







^^^^^lk 




ELEMENTOBUM LIBER m. 263 

fB, et z/F^ circulam jiBF secet, B^ autem con- 
ngat. dico, esse AJ X ^/F = JB*, 

recta^^F.^ igitur aut per centrum ducta est aut non 
er centrum. sit prius per centrum ducta^ et centrum 
TCuli.<dfjBFsit Z, et ducatur ZB. itaque LZSd rectus 
st [prop. XYnr]. et quoniam recta ^F in Z in duas 
artes aequales diuisa est^ et ei adiecta est FJy erit 

AJxJr+ ZT^ = ZJ^ [II, 6]. sed 

Zr— ZjB. quare 

AJ X ^r + ZB^ = Zz/«. 

est autem Z^* = ZB^ + Jl^/« [1,47]. 

itaque^^/X^r+Z5* = ZB« + Jlz/«. 

subtrahatur, quod commune est, ZjB^ 
»que AA X AT = ^M 

iam ne sit ATA per centrum ducta circuli ABTy 
t sumatur centrum £, et ab £ ad w<^r perpendicu- 
uris ducatur EZ, et ducantur EBy ETy EA. itaque 
.EBA rectus est [prop. XVIII]. et quoniam recta 
»er centrum ducta EZ rectam non per centrum duc- 

tam ^r ad rectos angulos secat, 
eadem eam in duas partes aequales 
secat [prop. III]. quare AZ = ZT. 
J^ et quoniam recta ^r in duas par- 
tes aequales secta est in Z puncto 
et ei adiecta est TA, erit 

.2. dV] in ras. m. 2 V. ZB] ZT P, corr. m. rec. 13. xA 
U]V', Ccov dl z6 Theon(BFVp). Ttfof iatl xa] P; TotgTheon 
BFVp). ^ 14. ZJB, Bd] z/JB, ZB P. Post Bz/ Theon add. 
i^^ yuq 71 vnh ZBJ (BVp et F, ubi d postea insertum est). 
20. Tol (pr.) m. 2 F. 22. EB] corr. ex EZ F. 23. 6ia] 
\ 6id B V. 26. tifivei,] (alt.) tsfisi: Bp. 26. ZF] in ra8. 

n. 2 V; rz F. 




r 




264 ETOIXEiaN y'. 

wxra v6 Z tfi^^ffov, ngoCxsiTai Si avry ij Pj 
aga vtio zav A^, ^T (isra tov axo i^g ZjT fffow 
iatl xa aao r^g Z^. xoivov nQoexEic&^ta tb dxo 
t^S ZE' t6 ttpa vno zmv AjJ^ ^T fitta xmv dno 
6 tav rZ.«Z£ teov ietl tols «lo ttov Z^, Z£. Eofg 
Si aitiy tmv TZ, ZE fffov iotl to awo Tijg ET' opfl^ 
yap [^ffrtj'] ^ vao EZT \y<avia]- totg ^^ dxb tcDV ^Z, 
ZE tcov iatl ro areo t^g E^' ro «pa tCTo rrov ^if^, 
AT n£ta tov ditb r^g ET ieov istl ta ditb tijs EA. 

10 ifijj Sl i] ET zfi EB- tb ap« vnb rav AA, ^T (is- 
ta tov dzo tijs EB fffow ietl xa «tio r^g EA. tca 
di «Bo tfjs EA tea ietl zd dxb tdv EB, B/i' opfrij 
ydq 71 vnb EBA ymvia- %b «p« usto Ttov A/i, ^T 
^std Tou «reo T]]s JEB ("eoi' iori ToFg d%o xmv EBf 

IG B^. xoivbv dq>'j]Q'^09e> tb djio tiis EB' Aotnov «po 
To vxb tmv AA, AT itSov i6tl rp «jro t^s jdB. 

'Edv aga xvxlov Xijipd'^ ti OtjfieCov ixtos, xal ctx' 
avtov Mpos Tov xvxiov jiQoUjtimoOi Svo ev^siai, xal 
17 filv avTiDV xd[ivy tbv xvxXov, i\ Ss itpantj^aiif 

20 ietat To vab oAjjs t^s TS(Jivover]s xaX tijs ^xrog djto- 
Ittfi^avofiivtjg fisra^v rov re ffijftftov xal r^s xvjiti 
asQKpsQgias feov ta dnb t-^s i<paiito(iivi]s tsrgayiav^. 
'AxsQ iSsi Ssi^^ai. 



1 



j '£«v xuxAov Xtiip&lj ti 6r]fislov ixtog, ajiH 
Sl roii erifieiov Kpos tov xvxXov xQosnCittaxn 
S^vo sv&siai, xal r) fthv avrdv tiftvy tbv xv- 

1. ajiiiciav} om. Bp. 2. ZT] TZ P. 4. iiJ] cort. in 

TO m. 1 B, ™ p, ^^] in ras. m. 2 V. 5. twv] (prins) «w 
F. roo»'] Pi faa BFVp. fotl'* F, aao tm»] inBert. m. I 



ELEMENTOBUM LIBER m. 266 

jiJx^r+ zr^ + zj^ [II, 6]. 

immime adiiciatar ZE\ quare 

jijxjr+ rz» + ZE^ — zj* + zei 

d EI^ — rz^ + Z^* [I, 47]; nam L EZF rectus 
t et EJ^ = z/Z« + ZE^ [id.]. itaque 

-^z/ X z/r + sr* = js^. 

d Br— JBA quare -^z/ X Jr+ EB^ = B^*. 
«i EB^ + ilz/* — jBz/* [I, 47]; nam ^ BBz/ rectus 
*. itaque JiJxJr+ EB^ — EB^ + ilz/l sub- 
«hatur, quod commune est, EB\ itaque 

jiJxJr—JB\ 

ESrgo si extra circulum punctum sumitur, et ab eo 
1 circulum adcidunt duae rectae, et altera harum 
rculum secat, altera contingit, rectangulum compre- 
ansum tota recta secanti et parte eius extrinsecus 
iter punctum et partem ambitus conuezam abscisa 
M|uale erit quadrato contingentis; quod erat demon- 
arandum. 

XXXVII. 

Si eztra circulum punctum sumitur, et ab eo ad 
irculum adcidunt duae rectae, et altera harum cir- 
ilum secat, altera adcidit tantum, et rectangulum 



ZJ] dZ P. TOftg W\ aUa xotg P. 6. FZ] P; z/Z F; 
d BVp. ^T] P; FE p m. 1; E^ BFV, p e corr. 7. 
^ ya^ — 8. T^s E ^] mg. p. 7. l<iziv\ P, om. B F Vp. £ Z F] 
ipra Fscr. ^ m. 2 V. ytavia] P; om. BFVp. z/Z] P; 
'Z BFVp. 8. iffi/] om. V. Ez/] P; TE BFVp. 9. 

15] F, TO 9. 10. EF] TE F. 11. i^rr/r» P, ut lin. 12. 
//] E corr. in ^ m. rec. F. 12. tcd»] ins. m. rec. F. 

I. yav^l m. 2 V. 17. 'ital a«* avTov — 22. Trr^y(»yo)] 

xl «i lj;^ff PBFV. 20. Tjjs 0X175 T^ff p. 24. X^' F.' 

7. xifi^si F, corr. m. 1. 



266 ETOIXEIfiN v'. 

xkov, »j di ZQoaitiztTj, jj ds To vzb [r^ff] oAt)s 
r^S t«fivoiJei;g xul r^g ixtbg ditolKii^avoftsvtis 
(iEta^v zov re ermtiov xa\ r^g xupT^s Kc^ti^iE- 
QEias tSov rra ajio t^s 3rpoOret:iroi;ffj;s, ^ JE^off- 
6 'xCn.TovCa ifpaiiistaL zov xvxXov. 

xvxXov yap roii ABF tlX^^tp&at ti ernistov ixTog 
To ^, xal «310 Toil iJ OTpog zbv ABF xvxXov ir^oa- 
Xiittitaaav Svo Bv&etai aC ^TA, dB, xal j] fiiv 
^FA tBftviroi tov xtJxAoi', ^ 5^ z/B xpoammEtQ}, ^eza 

10 5i ro ijro rtoji ^,^, AFlGov za anb t^q jdB. iiyeOf. 
oti ij ^B irpaxtstai. tov ABF xvxlov. 

"Hf^^at yuQ Toii ABT i<pantOfiivri ^ AE, xal sl- 
Xrj^Qa t6 xbvzqov rou ABF xvxXov, xal Satca t6 Z, 
xal inB%Evx&(o6av aC ZE, ZB, ZA. i] apa vno ZE^ 

15 6q&'^ istiv. xal iTtsl rj ^E i^paazEtai tot; ABF xv- 
xkov, rB(iVEt Si r) ATA, t6 aga vno rmv A/S, AT 
leov ietl ttn aab tfjg jdE. r)v 61 xal to v:r6 zav 
A^, ^r Heov za a%h t^s AB' t6 «p« KJt6 x^g ^dE 
leov iatl T^ anb r^j ^dB' Isi^ aQU ij ^E t^ jJB. 

20 iml. Se xal 71 ZE z^ ZB Her] ■ dvo dij aC ^E, EZ 
dvo zaTg /JB, BZ teai sieiv xal fideig avtmv KOivil 
Tj ZA' ■yavia aga ri vjtb iJEZ yavia zy waro ^BZ 
ietiv ter\, opfl^ S'b ^ vitb AEZ' op^ apor xal 7] vxo 
^BZ. xaC ieziv i] ZB ix^aXXoiiivt] didfiEtQos' 17 9i 

35 T^ dtafiBZQO) Toi5 xvxXov nffbg opdas dit' axQag dyo- 



i 



1. t^b] deleo; m. 2 V. 3l- in tas. m. 2 V. 2. inslH 

(priuB)PF,Vinrafl.,Bm.rec.! om. p. 6. xii«ioii] anpra m. 1 H 
P, 10. A^] A 7 -m. 1, Y ja. V, J anpra acr. FT m. B. ^ 

jr] r P; corr. m. teo. 13. Kevteov^P, F m. l, poat rae. 
V; Z ME«eD* Bp, F m, 2 (enan.}. nvvlov} m. 2 V. xccl 
iata, To Z] PFV; om. Bp. U. vno] rt vicoN, del. n m. 1. 
15. hti V. 17. Tfv Si *ai] P; uaoxttriw Sb Theon (BFVp). 




ELEMENTOBUM LIBER m. 267 

mprehensum tota recta secanti et parte eius extrin- 
cns inter punctom et partem ambitus conuexam 
»scisa aequale est quadrato adcidentis^ recta adcidens 
rculum continget. 
nam extra circulum ABF sumatur punctum ^, et 

a z/ ad circulum ABF adcidant 
duae rectae dFAy ^B, et ^FA 
circulum secet, ^B autem ad- 
cidat, et sit 

AJX^r^ JB\ 
dico, rectam JB circulum^£F 
contingere. 

ducatur enim circulum ABF contingens JE [prop. 
VUJ, et sumatur centrum circuli ABF, et sit Z, et 
cantur ZE, ZB^ ZJ. itaque L ZEA rectas est 
rop. Xyni]. et quoniam AE circulum ABT con- 
igit, secat autem JTAy erit AAx AT = JE^ 
rop. XXXVI]. erat autem etiam ^ziX^F= ziB*. 
.que ziJS* = zi5^; quare JE = JB. uerum etiam 
E = ZB, itaque duae rectae JEy EZ duabus JB, 
Z aequales sunt; et basis earum communis est ZJ. 
ique L JEZ = JBZ [I, 8]. uerum L ^EZ rectus 
b. quare etiam L ^BZ rectus; et ZB producta 
unetrus est; quae autem ad diametrum circuli in 

aQa\S\ aQcc, del. 9i m. 1 F. 20. iativ B. ZE] litt. Z 

ras. F. 21. Svai Vp. JB^BZ] corr. ex JE, EZ m. 2 

Bici Vp. 22. ZJ] litt. J in ras. m. 2 V. 23. tarj iatCv 

24. ZB] B, F post ras. 1 litt. (mg. m. 1: yq. ri JZ)\ 

S P, et V corr. ex ZB m. 2; EZB in ras. p. 



268 STOIXEIiiN y'. 

fidvri ig>cact€tai, tov xvxkov ^ ^B &fa ig>am£tai, 
tov ABF xvxXov. oiioimg Sri 8six^6€tai^, xav ro 
xivtfov inl trjg AF tvyxdvy. 

^Eav aga xvxkov ^fig^d^fj ti Crmstov ixtogy dxo dh 
5 tov ^nibBCov nffog tbv xvxXov nfo6nintm6i dvo ev- 
{^€tai, xal ^ (ihv avtmv tdiivt] tbv xvxXov, ri Sh nffo^- 
ximfiy y 8\ t6 vno okrig tijg t€(ivov6rig xal f^g 
ixtbg dnoXafiPavoiiiurig fistal^v roi) t€ 6ij(i€iov xal 
tiig xvQtfig n€Qvg)€Q€lag t6ov tp dni trjg ngo6mmov- 
10 6rigy 1} nQo6ni7Ctov6a iq>dil>€tav roi; xvxXov ' on€Q i8€i 
8€tiiaL. 



1. Tov] xov ABF Vp, F m. 2. xov %v%Xov' ri JB aga 
iipdmstai] mg. m. 1 B; item P, addito nai ante rov. fj /JB 
— 2. nvTiXov'] om. p; mg. m. 2 V. 2. ^ij] di V, corr. m. 2. 
3. AF] r ia ras. m. 1 B. rvyxavn P, corr. m. 1. 4. ano 
91 ^ 10. xvxXov] «al ta If^ff PBFVp. 11. JEvxAf^dov ac«£> 
Xsioav y PB, EviiXs^dov cxoi%sCoiv trii Oioavog inSocscag y F. 



ELEMENTOBUM UBEB m. 269 

nmno perpendicularis ducta est; circulum contingit 
)rop. XVI noQ.]. itaque ^B circulum .<^£Fcontin- 
ii similiter demonstrabitur, etiam si centrum in 
rcadit. 

Ergo si extra circulum punctum sumitur^ et ab 
I ad circulum adcidunt duae rectae, et altera harum 
rculum secaty altera adcidit tantum^ et rectangulum 
>mprehensum tota recta secanti et parte eius eztrin- 
»008 inter punctom et partem ambitus conuezam abs- 
sa aequale est quadrato adcidentis^ recta adcidens 
irculum continget; quod erat demonstrandum. 



a', ^x^fia fudi^jia^ftoi' et^ ii%i\ta sv^vyQafi- 
fiov iyYQafpea^ai Xiyezat, orav iKa6zri rav rov 
iyyqafpoiUvov ay^iq^aro; ycovtSv ixdatijs xi^vffag roii, 

6 tlg o iyyfidqsetai^ antijtai. 

fi'. ^XVP" ^^ 6(io(a}g WEpI ax^C-t «eQiyQd- 
rpBGifai liyatai, orav Bxderij nltVQK tow atQiyQa- 
qiofiivov exd€z^? ycovias Toi5, jrepl o rcepiypaqssTai, 

D y'. 2^xiip,a Ev&vyQa(i(iov bIs xvxXov iyyQatpe- 

e&ai Xiysrai, orav ixddr^ yiovia tov iyyQatpoydvm 

aTttTjtaL TTJs xov xvxkov naQi^SQEiag. 

S'. S^x^f^a S\ evQvyQafipiOv jcbqI xvxXqv 

QiyQd^ee&ai XiyetuL, otav ixderij «XsvQa roi 
% atQtyQK^ofidvov iipdittijtai rrjs tou xvxXov aiQUp. 

Qtias. 

e'. KvxXog Sl tig sy^^iLa 6fioia>e iyyQdipea&ai 

yettti, otav fj tov xvxXov neQKpdQsia ixdertjg ^iXEVQOi 

Tot!, tig iyyQaiperai, amrirai. 
) e'. KvxXog 6i ^sqI ffz^fia nsQiyQdipaadwi Xiystatf 

orav ij Tou xvxXov ntQiipBQeia ixdcrijg ymvCag i 

jttQl TtlQiyQatpetat, aTtri^tai. 



1. oeoi] om, BPp. 
post laa, 1 litt. V. 



NnmeroB om. PBP. i. y 

1. icEgiyfdipctai] inter t et y S lin 



IV. 

Definitiones. 

1. Figora rectilinea in figoram rectilineam inscribi 
dtar, com singnli angoli fignrae inscriptae singola 
«ra einS; in goam inscribitur^ tangont 

2. Similiter figura circom figuram circomscribi di- 
or, cnm singnla latera circumscriptae singulos an- 
los eiuSy circum quam circumscribitur^ tangunt 

3. Figura rectilinea in circulum inscribi dicitur, 
m singuli anguli inscriptae ambitum circuli tangunt. 

4. Figura autem rectilinea circum circulum circum- 
ribi didtur, cum singula latera circumscriptae am- 
Tom circuli continguni 

5. Similiter autem circulus in figuram inscribi di- 
m, cum ambitus circuli singula latera eius, in quam 
scribitur^ tangit. 

6. Circulus autem circum figuram circumscribi di- 
TOTy cum ambitus circuli singulos angulos eius, cir- 
.m quam circumscribitur^ tangit. 



Def. 1. Boetius p. 379, 19. 2. Boetius p. 379, 22. 

18. F. 11. imyQatpofiivov P. 15. i(pantritai] Bp; itp- 

;r£Ta» P; antrjtai FY. 17. di] dh ofio^g p. 6(io£cag] 

B; om. p; BvdvyQafifiov, supra scr. bfutimg m. 2, FV. 20. 
ijfia Bvd^vyQaitfiov FV. 



ETOIXEIiiN S'. 



za jciffaxa avxiqq ial trjg nEpupegtias 17 zov xvxi.ov. 



Elg tav So&dvxtt xvxkov xy do&E^at] EV&Eta 
6 jiij fiEi^ovt ovOt) xijs tov Kvxkov SiafJtixQOv 
tarjv EV&Etav dvaQfioSai. 

"Estio o ^o-&£(g xvxXos o ABF, ij di doQ^EtOa tii- 
&Ela (17) (isi^mv f^g Toij xvxXov Sia^ixqov i} ^l. Sel 
Si] eCg tbv ABF xvxXov t^ /i av&eia raiji/ ev&etav 
10 ivaffficeai. 

"Hx&to row jiBF xvxXov Sta^EtQog ij BF. eI piv 
orrv tSri daxlv i) BF x^ ^, yEyovos av eI'ij to iai- 
ta^piv iviJQftoexai yap elg rov ABT xvxXov r^ jd 
Ev&sia tarj fj BF. ei Si (lEi^cov icxlv ^ BF xr\s d, 
15 KEia^a XTj ^ taij.i] FE, xal xivt^ip t^ F SiaatJniaTi. 
Se t^ FE xrxAos ysyifa(p&(o 6 EAZ, xal litE%Evx&ca 

n FA. 

'Easl ovv xo r atjfiEtov xivtQov iatl rov EAZ 
xvxi,Ov, tarj iaxlv 57 FA tfj PE. alla t^ ^ fj FE 
EO iaxiv tOTi ■ xai i) ^d aqa xij FA iGxi,v tari. 

Eis «pa Tov So&ivxa xvxKov xov ABV tfj So- 
^eiati EvQ-Eitt iij ^ tenj iv^^^fioaxai ^ FA- otieq iSsi 
aoiijaai. 



i. 



Elg Toi' So&ivxa xvxXov Tra SoQivxt 
yavo) laoydvtov xgiyatvov iyyQafai. 

1. Boetins p. 3B8, 23. 11. Boetiua p. 3BB, B6. 

1. 1I5] e oorr. m. 2 P. ivagiio^ea^ai] Iv- m. 2 V. 

2. ItcI tijs TttQiiptQeiag ^ loiJ miiiioD] PBp. V mg. m. rec; 
ivft^uUj) T^ Tov xvxloti *(gc<pefeia F, V m. 1. 8. ffq] ^ J 



i 



ELElfENTOBUM LIBER lY. 273 

7. Recta in circalum aptari dicitur, cum termini 
eios in ambitu circuli sunt. 

I. 

In datam circalam datae rectae non maiori, quam 
est diametras circoli, aequalem rectam aptare. 

Sit datuB circulus ABF, data autem recta non 
maior diametro circuli sit ^. oportet igitur ixi ABF 
circulum rectae ^ aequalem rectam aptare. 

ducatur circuli ABF diametrus BF. iam si 

^ Br—j, 

effectum erit, quod propositum est; 
nam in circulum ABF rectae ^ 
aequalis aptata est£F. sia Br>^, 
ponatur rE= ^, et centro F, radio 
^ autem FE circulus describatur Eji Z, 
et ducatur FA. 

iam quoniam F punctum centrum est circuli EAZ, 
erit FA = FE. sed FE = ^. quare etiam ^ = FA, 
Ergo in datum circulum ABF datae rectae d ae* 
qualis aptata est FA'^ quod oportebat fieri. 

11. 

In datum circulum triangulum dato triangulo ae- 
quiangulum inscribere. 

fii) V. ri d] om. V; in F euan. 13. ivslgftoarai B. 
yap] supra m. 1 P. J] F\ B (p. 14. di] P, Campanus; 

dh ov Theon (BFp; d* ov V). 15. yLsMio] %al %B£<s»m Bp. 
^ivxgai \i.iv BVp. 16. EAZ] PF; in raa. m.2V; AZ Bp. 

18. EAZ] AEZ P. 19. r j z/] PP, Vm.2; 17 z/ Bp, Vm.l; 
d in ras. V. n FE] PF, V m. 2; t^ FE Bp, V m. 1; TE 
in ras. V. 20. z/] seq. raa. 1 litt. F.^ FA] AT FV. 
t6ri ictCv F. 22. rost svd^sia add. (iri fisl^ovi ovari xr^q xov 
%v%Xov diafiixQOv Bp, m. 2 mg. FV. ivsC^fioatai B. 

Euolides, edd. Heiberg et Menge. 18 




ETOIXEIilN i'. 



"Estai 6 So&elg xtixAog 6 ABF, ro d^ So&\v rpi 
yavov To ^EZ' SeI Sij eIs tov ABF xvxKov 
jd EZ xgiymvci Isoyoiviov TQiycovov iyyffaiinxi. 

"Hx&ai To£ ABF xvxkov ipaTtrofitvTj r} H& xaw 
f> tb A, xal ffuveffioToJrepos zi} A& sv&sC^ xal ta wpog 
ttVT^ 6ri(iEip Tca A t^ vnb ^EZ yavltf fffij ^ vKo 
&Ar, wpog 6% tfj AH t\i%ti(i xal t^ arpog avr^ 
erjfiaC^ ta A t^ vito ^ Z E [yavii>:] lei} tj vnbHA&, 
xaX inEtsvx&a i\ BF. 

10 'Exsl ovv xvxXov loii ABF ifpantBtai tig sv&eCa 
ij A&, xal anb tijs xata tb A ETtatp^g tig rbv xv- 
xXov Si^xtai Bv&sia rj AT, rj aga vnb &Ar t<ji} 
iatl T^ iv Tp ivttkka^ zov xvxXav tfttjfiati yavla t^ 
vTto ABT. aW 71 vitb ©ATt^ v%b JEZ isxJ iatf 

16 kbI j{ vxb ABF afftt yatvia t^ vjto AEZ istiv Catj. 
Sia ta avztt 6^ xal r] vnb AFB t^ vxb ^ZE isTiv 
ier}' xal Xoiitri orpK i; vao BAr Xouffj r^ vitb EjdZ 
ietiv fa») [iaoyoiviov ttga iezl tb ABF tgiyavov tm 
JEZ Tffiydv^, xttl iyyiypaittai bIs rov ABFxvxlov]. 

20 Eig tbv SoQivTK «pa xvxXov ta do&ivti tQiycSvat 
ieoytoviov TQiyavov iyyByganTai.- oaaQ sSsi ^iot^om 



^ 



/Tepl Tov So^ivta xvxAov rp do&iv 
vvq) ieoywviov tqiyavov XEffiyQailiai. 



in. . 



1. fltl m. ree. F. 3. zJEZ] Z poatea inBert. m, 1 F. 
4. H©] P (H in rM.), F, V m. 1; HAB Bp, V m. 3. 
JtDOtl «pte niv Bp. J@] H& F. 6. JEZ] iJ in ras. 
vno^ m. 2 F. 7. Jtgog Sc] jiaUv jipoc P. AH] HA P. 
,a,ria} om. P. 10. antCTat BT, 11. Ae]P; HAB F 






et V (ri i 



LB.]; 3^ Bp. Kttl ano] « 




ELEMENTOBUM LIBER IV. 275 

Sit datas circulos ABFy datns antem triangalas 
'£Z. oportet igitar in ABF circulnm triangulo 
'£Z aequiaiigQlum triangulum mscribere. 
ducafcur circulum ABF m A oontingens HB 

[niy 17]y et ad AS reetam et 
punctum eius A angulo ^EZ 
aequalis construatur L ^AFj et 
BdAH rectam et punctum eius 
A angulo AZE aBquBXisLHAB 
[I,23J, et ducatur BF. 
iam guoniam circulum^BF contingit recta A^y et 
» A puncto contactus in circulum producta est recta 
r, erit L SAF'^ ABF, qui in altemo segmento 
intas est [III, 32]. sed L^Ar=AEZ. quare 
iam LABF^ AEZ. eadem de causa etiam 

L ATB — AZE. 
ique etiam LBAF^ EAZ [I, 32]. itaque trian- 
ilo8 ABr aequiangulus est triangulo AEZ^ et in 
rculum ABF inscriptus esi 

Ergo in datum circulum dato triangulo aequian- 
dus triangulus inscriptus est; quod oportebat fieri. 

ni. 

Circum datum circulum dato triangulo aequian- 
ilum triangulum circumscribere. 

A inatprig stg tov xvxXov] aqp^g Bp. 12. sv^sia] tig Bp. 
Post SAF m B ins. y(ov£a m. rec. 14. dXXd P. 15. 

a yav£a] in ras. m. 2 V; yoivCa dga F. JEZ] litt. JE 

ras. m. 2 Y. 16. did ta avxd — 17. Ccri] mg. m. 1 F. 
.ATB] FB e corr. m. 1 p. JZE] E in raa. m. 2 V. 17. 

wjl m. 2 V. EJZ] E ins. m. 1 p; JEZ F. 18. ftny 

r^^^Fp. lcoymviov — 19. ytvnXoviom. P. 21. iaoya' 

9 F; corr. m. 1. noiTJGai] Ssi^ai. B Y; iv aXXm* de^Sat m. 
mg. F. 

18* 




276 ETOIXEIiiN e\ 

"Earca 6 SoQ^sls xvxlos o ABF, ro d\ 
yavov t6 ^EZ- Set Si} Jtspt rov ABF xvxXov tp 
i/£Z T^tydva' isoycovtov XQiyeavov itSQiyQiitliai. 

'Ex^E^X^e&ea 7\ EZ S<p' ixatcQa ta /i^pij xata 
^ Ta H, & Utfnita, xal e/AijgiS-ra toii ABFxvxlov XEWpov 
tb K, xal Sf^x9<ii, mg itvxtv, £v#£fa ^ KB, xal gvvs- 
atata n;p6s rjj KB sv&ai^ xal ta zqos avt^ 6t](ttim 
Tc3 K tfi niv vao jdEH yavia t9i\ rj vnh BKA, rjj 
Si v«6 JZ® teri ij vno BKF, xal d,a tmv A, B, F 

10 erjfisiav ^x^raffav ^gjarcrojtErai rov ABV xvx?.Ov ai. 
AAM, MBN, NFA. 

Kal ijtEi iipanzovtai tov ABF xvxXov aC AM, 
MN, NA xata ta A, B, F arfiiita, ano Se roiS K 
xivTQOv inl ta A, B, F STjfieta iitE^tvyjiivai slalv 

15 of KA, KB, KF, op*ai apa siaXv aC TtQog totg A, B, 
r aijitsiois ymviai. xal sjtel tov AMBK tttQanXsv- 
Qov a( TEffffapfg yayviai thgaaiv OQ&aig Caai siaiv, 
dasid^itsQ xal bIs Svo zpiyava SiaiQetztti t6 AMBK^ 
xai eiaiv oQ^al aC vno KAM, KBM ymviai., Aoincel 

20 uQa a( V7t6 AKB, AMB dvaiv 6(}&atg taai elaiv. 
eial Sl xal aC vito AEH, dEZ Svalv opdar<; taaf 
al aQa vito AKB, AMB tats vKo dEH, AEZ 
taat tlaiv, mv ri vx6 AKB t?; uko ^EH iattv tar] 
XoiJti} aga ij vito AMB Xomfj ty vao ^EZ ian 

26 fffjj, ofioitos Sij Seix^rjettai , oti xal ij vao AN^ 

1. ie] om. p, Bupra, F. 4. xoia] PBFpi l-ni V, 

H, 0] in ras. P; H in ras. m. 2 V, S. KB]BK F. 

SKA\ litt. KA in raa. m. 2 V. 9. faij] m. 2 V, 13. MNm 
N add. m. 2 poat ra». V. N A\ A add. ra. 2 post ros. V "^ 
oij/ijftt] Bupra P; om. Bp. arto B^k io« — 14. flijfititf] 

P. 14. *ji£feyyfie»oi] P; i-itii,evyvviifvat BFVp, 19, . 
lieiv (Jfl&ni] P; «ipajrlftieoi', (o* Tfieon (BFV; corr. ex i 
ipctyoivov uv ra. 1 p). Hf] supra m. 1 P, MAK P. 



1 




ELEBfENTOEUM LIBER IV. 277 

Sit datus circulns ABFy datus autem triangulus 
dEZ] 0{K>rtet igitur circum ^BFcirculum triangulo 
dEZ aequiangulum triangulum circumscribere. 

educatur EZ in utramque partem ad puncta Hy 
9j et sumatur K centrum circuli ABFf et producatur 
itcunque recta KBy et ad rectam KB et punctum 
dus K angulo dEH aequalis construatur L BKA^ 

angulo autemz/Zd aequalis 
Z.Biirr[I,23]. et perpuncta 
Ay B, r ducantur circulum 
ABF contingentes AAM, 
^ MBN, NFA [m, 17]. et 
^ quoniam AMy MN, NA 

drculum ABF contingunt in punctis A, B, F et 
i centro K ad puncta Ay By F ductae sunt KA, 
KBy KFy anguli ad Ay B, F puncta positi recti sunt 
llli 18]. et quoniam quadrilateri AMBK quattuor 
inguli quattuor rectis aequales sunt^ quoniam AMBK 
in duos triangulos diuiditur [cfr. I, 32], et anguli 
KAM, KBM recti sunt, reliqui AKB + AMB duo- 
i)us rectis aequales sunt. uerum etiam AEH ^ AEZ 
duobus rectis aequales sunt [I, 13]. itaque 
AKB + AMB = AEH + AEZ, 
juorum L AKB = AEH quare L AMB = AEZ. 
dmiliter demonstrabimus, esse etiam L ^NB = AZE, 



^mvCav] P; ytovCai $vo oq^cci bUiv B et p {bIci)\ ymvCai. 
Ho OQ^aig taai. staiv F et V {dvaCv et stai), XoinaC 

- 20. elaCv] bis F. 50. slatv Caat p. 21. slaC] bIoCv P. 
M di — taaC] mg. m. 2 V. 23. taai sCaCv, mv 17 vno] in 

•as. m. 1 B. 26. 8-^] Si F (corr. m. 1), V (corr. m. 2). 
iNB] Bp; FNB P; ANM V (N corr. ex H); ANB F seq. 
ipatio 2 litt.; A corr. m. 2 ex A. 



r 



278 ZTODCEIiiN e'. 

TJj vao jJZE ietiv iaif naX AotjtiJ apa ^ uko MAN 
[Aotw^] XTj vno E^Z iaxiv iarj. i&oycaviov ap« iszl 
v6 AMN Tffiymvov %a AEZ zQiywva' xal Tcsgiyi- 
yfftacTcii irepl tov ABV xvxXov. 

JIsqI tov SS^ivTcc aQa xvxkov Tto do&ivri Tpt- 
ymva Isoyaviov z^iyavov nsQiyiyQamui " oXBp idsi 
xot^eat. 

d\ 

Eig To So&lv TQCycovov xvxXov iyyQaijiai. 

"EeTa ro So&lv tQCytavov ro ABF- Sit dfl Btg to 
ABF TQfytavov xvxXov iyyQail>ai. 

TsTfi^g&aeav a{ vao ABT, AVB yaviai W/ff 
T«rs Bz/, FA fv^eiaig, xal Gvft^alXhaeav otli.^^Xais 
xara x6 A etjfiBlov, xal ^^&meav axh xov jd inl xaq 
AB, Br, rA Ev&Eiag xaa^erot aC AE, AZ, AH. 

Kal ixBl fuj) iaTlv ij vito AB^ yavCa x^ vxo 
FBA, ietl S% xal 6q9^ ]j vao BEA op*J7 r^ vxo 
BZA Hat}, dvo 8r] XQCyava ieti xa EBA, ZB^ tas 
Svo yavCas xatg Sval yavCatg teag i^ovra xai fiCttv 
jtXevQav ftta xXBVQa l^etjv njv vnotBCvoveav vao fiCav 
■tmv ta<av yavtav xoiv^v avtav xijv BA- xal xicq 
lomag aQu nXEVQag xais kotnatg xkBVQatg teas s^ov- 
<Hv' fffij a^a ij AE xij AZ. Sta r« avra Sjj xal tj 
^H xy AZ ietiv ter]. aC xQBtg fipc sv&etat <d ^ 



IN ■ 

9zi m 

I 



rV. PappuB VII p. 646, 7. BoeiiuH p. 389, 1 



% 



1. JZEl .ilEZ F. 2. Xainri] otn. P; fmvia lotit^ FV. 
EJZ\ aEZ F. eW» P. 12. AFB] PP, V m. 2; BH* 

Bp, V m. 1. 13. evtt^alXiteKiav^ alt 1 eupra rn. 1 F. 

15, rA\ A in raa. p, corr. es J B, 18. A&A] B in ras. P. 
17. TB J] rjB, corr. m, 2 in JBZ P, «rfiijToi yaf Stpx 

mg. p, iaiiv B. 18. ^ort] lativ P; t^otV. ZBJ] PF, 



dtt 



ELEMENTOBUM LIBER IV. 279 

qualre etiam LMAN=EdZ. itaque triangulus 

AMN tiiangulo AEZ aequiangulus est; et circum 

ABr circulum circumscriptus est. 

Ergo circum datam circulum dato triangulo ae- 

quiangnlus triangulus circumscriptus est; quod opor- 

tebat fieri. 

IV. 

In datum triangulum circulum inscribere. 

Sit datns triangulus ABF, oportet igitur in trian- 

gulum ABF circulum inscribere. 

secentur enim anguli ABF, 

AFB in duas partes aequales 

rectis B^, Fjdi [I, 9], quae con- 

currant in ^ puncto [I ah. 5], 

et a ^ ad rectas ABj BF^ FA 

\jj perpendiculares ducantur AE^ 

AZy AH, et quoniam 

LABJ = rBJ, 

ei LBEA = BZA, quia recti sunt, duo trianguli 

EBjd, ZBA duos angulos duobus angalis aequales 

habent, et unum latus uni lateri aequale, quod sub 

altero aequalium angulorum subtendit commune utrius- 

que£^. itaque etiam reliqua latera reliquis lateribus 

aequalia habebunt [I; 26]. itaque AE = z/Z. eadem 

de causa etiam AH=AZ^) ergo tres rectae AE^ 

AZ, AH inter se aequales sunt. itaque qui centro 

1) Nam /, z/rif = z/rz, JHr = jzr, jr=^jr; 

tam u. 1,126. 

ixovTsg V, corr. m. 2. 20. rrjv] om. Bp. 24. rj] secj. ras. 
1 litt. B. Post l'6ri add. Theon: oactB xal rj JE rw JH 

iativ tcri (BFp et om. laxiv V); om. P, Campanus. at tgsig 
— 280,1: dXXrjXaig slch] om. p; mg. m. rec. B. ev&8iai]om.y. 




ETOIXEIiiN e\ 



^Z, jdH ieai aXX^Xaig iieCv 6 «Qa xdvTQa xa 

. Mol diaOz^iiati ivl rav E, Z, H xvxXog ypa^oiiBvoi 

ij^ei xal dia rtov Xoixiav eijftFiav xal i<pai{iiTai rt 

jiB, Br, FA ivQsimv Sia t6 op4^ag iivai rag npo^ 

b roig E, Z, H Sri(i£(ois yoiviaq, ti yuQ rsiist avzdg, 

ietai ^ rfj Siafihga tov xvxXov Jtpos OQ&ics iaC 

ax(fKg ayoytivtj ivros nintovea tov xvxXov' oaBQ ato- 

3tov tdeix^' ovx a^a 6 st^i^pc) rp ^ SiaerijfiaTi 6h 

ivl xmv E, Z, H Ypa<p6fi£vog xvjc/os TSfiBt rag j4B, 

10 BV, FA \{v&tCag' [iipd^iiBtaL aqa avTmv, xal letat 

MvxXos iyyfyffafifiivog sig t6 ABF TgCycavov. iyyg' 

ypatp&to ag o ZHE. 

Eig ftpa t6 So9\v tQCyavov to ABFxvxXo^ ^YV^. 
ypantai 6 EZ}H- otibq ISbi xot,^6ai.. 

16 a'. 

HsqI to 6o&'h'y tQCycavov xvxXov neQt- 
yQa^fai. 

"Eetoj t6 tfo^fv tpCyavov t6 ABF' Sbi 3e «*) 
T(. Sod^lv TQCytovov t6 ABT xvxXov ntQtyQa^ai. 

20 rttfiiJiTS-citfer' al AB, AT Ev&Etai SC^a xara n 

jj, E et]fi,eta, xal kbo tcov ^J, E erjfitCav tatg A\ 
AF Wpos op^ag ^xQcoaav aC /iZ, EZ' avn%B6ovvTi 
Sif ^TOt ^VTog lov ABF TQiytovov ij iitl t^g BP ei 
&ECas ^ ^3£t6s tijg BF. 



1 



V. Pappna '¥11 p. 648, 7. SimpliciDB in pbjs. fol. H». 

1. fattil tv»fiai iitai V. cleC V. 2. mb^] m. 2 V. 
1«'] ei Ivi V et m. rec. B. E, Z, H] PBp; JH, dZ, A. 
in raa. V et, ut niiietnr, F; jp. %ai- tiaX iv\ zmv JH, JZ, jJ, 
rog. m. tec. B, ypoftpofifjUjvos P, 6. yoivittg] m, 2 V. 

tip-n B, 6. ««'1 litt, a- in raa. m. 2 V. 7, oir*e ketiv Vl 
8. iS"'z«ij] P, B m, rec; om. Vp; xnl Met'iff^ F. o] — ' 



ELEMENTORUM LIBEB IV. 281 

3t radio qualibet rectarum ^E, ^Z^ ^Hy) descri- 
ir ciroulu8| etiam per reliqua puncta ueniet et rectaB 
!, £Py TA contingety quia recti sunt anguli ad 
icta E^ Z, H positi. nam si eas seca^ recta ad 
netrum circuli in termino perpendicularis ducta 
a oirculum cadet; quod demonstratum est absur- 
1 esse [III; 16]. itaque circulus centro zf et radio 
libet rectarum JE^ JZ^ JH descriptus rectas AB^ 
\ FA non secabit. itaque eas continget; et circulus 
rianguIum^BFinscriptuserit. inscribatur xxtZHE. 
Ergo in datum triangulum ABF circulus inscriptus 
BZH] quod oportebat fieri. 

V. 

Circum datum triangulum circulum circumscribere. 
Sit datus triangulus ABF, oportet igitur circum 
um triangulum ABF circulum circumscribere. 
secentur rectae AB^ AF in duas partes aequales 
punctis A, E [I^ 10]; et a punctis A, E 2Li AB, 
^ perpendiculares ducantur AZy EZ. concurrent 
iur aut intra triangulum ABF aut in recta BF aut 
•a BR 



1) Graecam locntionem satis miram et negligentem sae- 
i (p. 280, 9. 282, 8. 290, 22. 292, 3) praebent boni codd., 
m ut corrigere andeam. 

S, Z, H] PBFVp, ed. Basil.; JE, JZ, JH Gregorius. 
vnlog P. TSfisi:] PV, F m. 2; tifivH Bp, F m. 1. 10. 
I rj e corr. m. 2 V. 6] om. Bp. 11. iyysypaqp^o» ms 
,HE] P; om. Theon (BF Vp). 13. sis] oc poet ras. 2 litt 
corr. m. 1. Sod^evti P, corr. m. 1. yiyqantai F. 

6] om. P. 20. AB] BA P. Ta] to F, sed corr. 22. 
] ^ e corr. P; ^r fv^iiaig F m. rec. EZ] ZE P. 
9ri] P; Sb BFVp. ^] supra m. 1 F. 




282 STOlXELntf 6'. 

Zv^mntitfneav x^otBQOv ^vzog xaza vi , 
imeitvx&iaeav at ZB, ZF, ZA. xaX iitiX FOt] ietlv ^ 
A^ zfj ^B, xoivi] de xai itpog opd-ag ^ ^Z, fiaats 
aga ii AZ ^ixSei tfj ZB iffttv leti. ofioims Si] dEi^Ofttv, 
5 ott xal ri rZ Tj; AZ iotiv fffjj' raffrs xal 17 ZB 
tfj Zr idtiv fffij- at TpEtg «pa at ZA, ZB, ZT fff« 
alXi^Xais siaCv. 6 apa xsvtpct ta Z Sta6f^nati di 
ivl tav j4, B, r xvxlos yQaqiofitvos i]S,u xal ^ia 
rtov XoiTtav atijiEiav, xaX istab iCEQLyEyQafiiiivos 

10 xvkIos JtepX to ABT tpiymvov. uitQiytYQatp^ rag ij 
ABF. 

dli.a di) at dZ, EZ evfiTitTttitaeav iffX vijg BT 
6v&etas xata to Z, cog ix^t iitl tijg devrigas Xttta- 
yQa<p^g, xal ine^Evx^oi 15 AZ. oftoimg ^^ SEi^ofiEV, 

16 oEt To Z atjjiEtov xivtQov iezX zoti stfpl to ABV tffi* 
yavov ^iEQtyQaipoiiivov xvxlov. 

'AXka Si] at AZ, EZ m'ftatxtitGieav ixtos i 
ABF tQiytovov xaza ro Z naktv, rag Ix^t i%l r^g 
rpiiijg xatayQatptiS, xaX iKEt^EVX&atGav al AZ, BZ, 

20 rZ. xal ijtel nakiv iffij ^ffTiv ij Ajd rij jdB, xotvi^ 
3% xaX XQts oQ&as ^ ^Z, Paeig «Qa fj AX pdeEt i 
BZ iettv tat}. o^otrag Si] Ssiiofiev, ort xaX ■^ TZ 4 

1. evnjriimnaav F. n^otfgov iycos] ovv Ivtos npOMpo* 
P, a. Zr] litt. Z in raa. m. 2 V, in T mutat, m. 2 P. 
3. zfBl BJ P. JZ] ^Z? F. 4. ZB] in raa. p. ^oeic 

foij] PF; roTj iortV BVp. B. rz] Zr Bp, 6. ieziv} om. 
T, Post foij raa. 6 litt, F. 8, -4, B, T] P; Z^, ZB, ZT 
Theon (BFVp), «cil Sia imv loixmv Bt]p.t/aiv] om. p; mg. 

m. rec. B, 9, 0] ineert. m, 1 V, 10, xal neQiygaipia^io 

V: «ai etiam ia F add. m. 2 (euan.), 13, BT] yJF F; conr. 
m. 2, 14. AZ] Z in raB. p. 19. AZ] AZ F. 
P; ^Z,rZ F; ZB.ZT BVp, !0, itai] eras. V 
PF, V m. 1; ZB Bp, V m, 2. FZ] ZF P. 



:l 

ra- 
lev, 

^''^ 

BZ, 




ELElfEMTORUM LIBEB lY. 288 

priiis igitur intra concurrant in Z, et ducahtar 
, Zr, ZA. et quoniam A^ ^ JBj communis 
nn et perpendicolariB AZj erit AZ = ZB [I, 4]. 
iliter demonstrabimus, esse etiam FZ ma AZ^ quare 
m ZB » Zr, ergo tres rectae ZA, ZB, ZFinter 
leqoales snnt. itaqne qui centro Z et radio qua- 
b rectarum ZA, ZB, ZF describitur circulus, etiam 
reliqua puncta ueniet et erit circum triangulum 
r circumscriptus. circumscribatur ut ABF, 






im uero AZ, EZ in recta BF concurrant in Z, 
t factum est in figura altera, et ducatur AZ, si- 
ter demonstrabimus, punctum Z centrum esse cir* 

circum triangulum ABF circumscripti.^) 
im uero AZ, EZ ultra triangulum ABF concur- 
t;') in Z, sicut factum est in figura tertia, et du- 
bur AZ, BZ, FZ, et quoniam rursus AA = AB, 
dZ communis est et perpendicularis, erit [I, 4] 

= BZ. similiter demonstrabimus, esse etiam 

rZ = AZ. 

1) Htinc casnm segregauit Euclides, quia hic sola A Z du- 
ia est. 

2) Quamquam offensionis non nihil habet inconstantia) qua 
lo i%to^ xov ABT xQiymvov (p. 282,17. 284, 15) scribitur 
lo Jxtoff T17S BT (p. 280,24), iamen tiJs BF contra codices 
80, 24 nix cum Gregorio in xov ABF xqiymvov corrigendnm 
^p. 282 , 15 iam ex r correctum est) , cum optime intellegi 
at| modo knxog uertamus: ultra. 



284 STOIXEiaN S'. 

j4Z ietiv ieri' mffrf xal tj BZ rfj ZF iilTiv l'Si)' o 
aga [miXiv] xfVrpra rfS Z Staaf^fittTi dk ivl xav 
Zj4, ZS, zr xvxlos ypayDftEvog ^Jet xal dia tav koi- 
nmv erjfiBiiav, xal ^Srai neQiyByQafi^ivo^ itB^l t6 ABF 
5 Tffiymiov. 

IIeqI ri io^ii' Kp« TQiyaivov xvxlog nepiyiyQaarai- 
OTttp i'd£i. JCOlfjtSai. ^fl 

Ktil <pavi^6v, OTt, OTB (ihv ivrog tov tQiycavov 

10 jtintti t6 xivTpov zov xvxkov, ^ vnb BAP yavltt iv 

liBitovt ifi^fuCTC lov TjiiixvxXiov tvyxttvovea iXartiav 

iarlv opQ^g' ote Si inl tijg BF ev&slas ro xivtffov 

xixtBi, ij vnh BAF yavia iv 7i(iixvxiia tvyxavovoa 

opftij ietiv ■ ots Sl t6 xivxqov tov xvxXov ixtog 

15 roti rpiydvov ninzBi, ri vjto BAF iv ikdrtovi rft?;'- 

^ati loi ij^txiixAi^ou ri^j');'* '''"'•''' fi£/£(Di' iatlv OQ^^g. 

[SetE xttl otttv iKttTTejv op&^s tvyxavr\ i) SiSoptivji 

ymvia, ivtog roiJ tQtycavov ntOovvTai aC AZ, EZ, 

oiKv 6i op&ij, inl Tijs BF, otav dl (iBi^tDv opO^^, 

20 ixTog i^s BF' otcsq IStt. jrot^ffat.] 



rof So&ivttt 



VI. Boelina p. 3 



■ ], JZ) in raa. m. 2 V, EZJ ZB ?. ZT] FZ BPj 

PoBt Fir;) m F iiiHert. in raa. at TQfis apu fcai uXl-^lais tter 
idem h tog. ta, rec. 2, naUv] om. P. 6. Poat le/yan 
Theon add. «ifiyivfBip^mattiABr (BFVp; yeyfaip&oiF a 
p; Koi /fyeiiijP&ai V, F m. 2; ij ABF F, corr. m. 2). 8. ; 



'" 



ELEMENTORUM UBER IV. 285 

re etiam BZ ^ ZF. itaque qui centro Z et radio 
ibet rectamm ZA^ ZB, ZF describitur circulus; 
n per reliqua puncta ueniet; et circum triangulum 
r circumscriptus erit. 

Srgo circum datum triangulum circulus circum- 
)tuB est; quod oportebat fieri. 
Bt adparety si centrum circuli intra triangulum 
ierit, angulum BAF in segmento maiore, quam 
semicirculus, positum minorem esse recto, sin cen- 
1 in recta BF ceciderit, angulum BAF in semi- 
nlo positum rectum esse, sin centrum circuli ultra 
igulum ceciderit, angulum BAF in segmento 
ore, quam est semicirculus, positum maiorem esse 
o^ pn, 31]. 

VI. 
In datum circulum quadratum inscribere. 



1) Finem (lin. 17 — 20) genninnm esse uix patanerim; pa- 
enim necessarins uidetur, et rj SidotAsvrj ymvia lin. 17 fal- 
est, ut obseruauit Simsonus p. 353, cui obsecuti locum 
Lgere conati sunt Gregorius et Augustus. haec uerba ideo 
lue suspecta sunt, quod speciem corollarii efficiunt, cum 
en uerba lin. 9 sqq. non corollarium sint, sed additio ei 
lis, quam in III, 25 inuenimus ; nam neque in optimis codd. 
nm noQiafia habent, neque a Proclo ut corollarium agnosci 
ntur (u. ad IV, 15 noQiGficc), 

a] om. P; mg. m. 2 BF; mg. m. 1 Vp. 9. oti, otb] otav 
10. n^ntsi] nintrj F; nCntoi P. yanvia'] m. 2 V. 12. 
:£at — 13. ymvCa] P; om. Theon (BF Vp). ^ 14. kotw] P, 
ipra m. 1; ^<rrat BVp. to nivtQOv tov xvxXov] P; om. 

on (BFVp). 15. tov tQiymvov] August; tQiYmvov P; tijg 
Bvd^sCag to nivtQOv BV^p; tov BF to 'nivtQov, postea addito 
B^g et tov in tijg mutato m. 2 F. nintff F. Post 

r in BFp add. yonvia; idem V m. 2. 18. tov] om. F. 
wvtai] P; Gvfineaovvtai BVp, et F, sed dei. avft-, 20. 
fffai] PF; dfitjat BVp; yp. SsC^ai mg. m. 1 F. 




286 STOIXEIiiN &■. 

ARFid xvxXov TET^dyaivov iyyQKiliai. 

"Hx&^afav totJ ABF^ xvxXov Svo diafiETQOi 
OQ^as aXl^Xttis «^ -^r", B^, xal ijutsvx&aOav a[ AS 

5 sr, rj, JA. 

Kal inel fffij ierlv ^ BE tfi EA' xivzQov ye^ 
ro E' xoivii 8i xal srpos o^Q-as ij EA, /Safftg a^a 
1] AB fiuSsi tij AA iOtj ietCv. Siu xa uvta Sij xal 
ixexiQa tav BF, FJ ixateQoc rav AB, AJ tOij iatCv 

10 CeoTiXEvgov a^a ietl t6 ABFA rftQan^evpov. X^a 
tf^, oTt xal 6(f&oyiavi,ov. ixel yap ij BA sv&srK Sia- 
yttTQOs iari tot) ABFA xvxXov, rnitxvxUov aga ietl 
To BAA' OQQii apa tj vico BAA yavCa. Sia t« 
avta Si} xal ixaQtr} rmv vroo ABF, BFA, FAA opS^ 

15 ietLV o^%oy(oviov ttQK ierl t6 ABTA rstQaJiXBVQOvM 
iSsix^ Si xal ieoaXtvQov " rttQuytovov &Qtt iext'» 
xal iyyiyQUJttai els titv ABVA xvxXov. 

Elg uga roi/ do&ivta xvxXov terguyavov iyyiyQoi 
rai to ABFjJ' owep iSst itoiijeai. 



IIsqI tov So&ivttt xvxkov rsrQdymvov niQt^ 
yQKiiiai. 

"Eero3 b So9Elq xvxXoq 6 ABFA' Ssi Sri jrrpi x{ 

ABFA xvxlov rerpttyavov TtsQiyQtt^ai. 

i "Hx^aaav tow ABPA xvxKov Svo SidiistQOi npj 

6Q9as dXX-^Xttis at AF, BA, xal Sia rav A, 3, F, 

3. i} ^jfriDirai' p. roC] yie rov Bp; eIs tov F. i 

kIov F. avo] om. BVp. 5, ^A] corr. ex FA ta. l F. 
7, Bpn] om. Bp. 8. tei(v1 F; comp. p; ieri PVB, 

^ori'!' P, comp. p. 12. iaTi] ietCv P. 13. yujv/u] e 
16. leriv] P, comp. p; iori BPV. 19. «pa] om. V. 




ELEBfENTOBUM LIBER IV. 287 

3it didiiip circulus ABFjdi. oportet igitur in circulum 
r^ quadratum inscribere. ^ 

luean^ar circuli ABFj^ duae diametri iuter se 
leudiculares jiF, Bjd, et ducantur AB, BF^ FA, 

et quoniam BE = EA (nam E cen- 
trum est); et EA communis est et per- 
pendicularisy erit AB = AA p[, 4]. ea* 
^dem de causa BF = AB et FA = AA. 
itaque quadrilaterum ABFA aequilate- 
rum est. dico, idem rectangulum esse. 
nam quoniam recta BA diametrus 
circuli ABFA, semicirculus est BAA, itaque 
AA rectus est [III; 31]. eadem de causa etiam 
^ili anguli ABF, BFA, FAA recti sunt. itaque 
angulum est quadrilaterum ABFA. sed demon- 
tum est; idem aequilaterum esse. itaque quadra- 
. est [I def. 22]. et in circulum ABFA inscrip- 
L est. 

Ergo in datum circulum quadratum inscriptum est 
rA\ quod oportebat fieri. 

VII. 

Circum datum circulum quadratum circumscribere, 
Sit datus circulus ABFA, oportet igitur circum 
\rA circulum quadratum circumscribere. 
ducantur circuli ABFA duae diametri inter se per- 
diculares Ar, BA, et per A, B, F, A puncta du- 

ta] ABTJ Bp; So&svta aga V. Post xvxXov add. tov 

rj V et F m. 2. 19. noirjcai] in ras. p. 24. tstQd- 

OQOv P. 25. yaQ tov Bp. dvo] om. p. 26. at] om. P. 



288 ETOIXEIiiN *'. 

»Jr}ftei<ov ^x&iaeav itpanTOfifVtti rov ABT^ xvxkov ccfS 
ZH, H&, ®K, KZ. V 

'End ovv itpaTcxatKi. f] ZH tov ABF/d xuxAoi^l 
«reo Sl Tov E xBVTQov inl ttiv xaza z6 A iitatprfv 
5 ini^evxTUi ij EA, a£ uqk Jipog rt5 A yaiviai, ogf^ai 
eCaiv. Sitt Ttt ttVTtt dij xal at %qoq Tolq B, T", ^ 
Ori^Biots ytaviai oQ&ai eisiv. xal inal op*ij iativ ^ 
vno AEB yavia, istl di opS-^ xal tj vno EBH, 
xaQal^.^Xos aga i^tlv ij H& zfi AF. Sta rit avta 

10 Si} xal 7] AF tTJ ZK ieti 3tapd)i?.f]7.os. Sats xal fj 
H& TTi ZK ioTi TCtt^dXXrjXos. 6(ioias Sij Sti^ofitv, 
oti xal ixaTi^a rmv HZ, &K rjj BEA iert aaQai 
XrjXog. TtttQalXijXoyQttfifue apa ietl ta HK, Hr„ A. 
ZB, BK- l'<f7} a^a iatlv rj fiiv HZ tf &K. 

15 H& ry ZK. xttl i:iel te^ ioTlv ij AF x^ BJ, dXXa 
xal 7) nlv Ar ExatiQa Tmv H®, ZK, i] Si B.d ixa- 
riQtt Tmv HZ, ®K ieriv [aij [xal ixatiqa a^a tav 
H@, ZK ixariQtt tav HZ, &K iCTiv fOij], {aojtXsvQov 
UQtt iatl to ZH@K tBtQdxXsvQov. Xiya S'^, OTt 

ao xal oQ^oytoviov. iatl yuQ itttpttXXtiXoygafiiiov iatt 
to HBEA, xai iativ op&^ rj vno AEB, dp#^ apa 
xal i) vjto AHB. ofioimg Sij Sei^ofiiv, ort xal at 
jtQOS Tols &, K, Z yaviat oQ&ai iiaiv. oQ&oyaviov . 
ttQtt iatl To ZH&K. iSeix&rj Sl xal laoTtXevQi 



i 



S. KZ) in raa. P; mutat. in ZJr in. 2 V. *. ^nnqsqi 

imqiaveiar p et B m. 1 (corr. m. rec). 5. na^ ' " 
fiat BVp. 7. titi Vp. 8. ABS] B in raa'. ' 

B in ras. F. 10. Tta^aUijlos iettv T. mon 

dUjjXoe] Pp (in ZK litt. Z in raa. p); om, Vj ^ _ 

m. S B; habet Campanus. 13. PoBt itaQuXliilos add. avte 

Moci Ji HZ T^ 9K ioTi naiidX3.Tilos Fp, B m, rec. HK] eras. 
P. 14. ZBJ in rae. F; B e corr. m. 3 V. BK] in ras. F. 
15. alla %ai] P; bU* BFVp. IG. ZK] ZK lerir faTj 



i 



ELEBIENTOBUM LIBEB IV. 289 

ttor cirealam ABFJ contingentes ZHy HS, BK, 

: [in, 17]. 

iam qnoniam ZH circulum ABTd contingit, et 
E centro ad punctum contactus A ducta est EA, 
^ ad A positi recti sunt [III, 18]. eadem de 
sa anguli ad puncta By Fy A positi recti suni et 
)niam L AEB rectus est, et L EBH et ipse rectus, 
', HB rectae AF parallela [I; 29]. eadem de causa 
un AF rectae ZK parallela esi quare etiam HB 
bae ZK parallela est [I^ 30]. similiter demonstra- 
mSy etiam utramque HZ, BK rectae BE^ paral- 

lelam esse. itaque parallelogramma sunt 

HK, HF, AK, ZB,BK. itaque [I, 34] 
HZ = BK, HB ^ ZK. 

et quoniam AF = BA, et 
AF^HB^^^ZK 

eiBA = HZ^BK [1,34], aequilate- 
Q est quadrilaterum ZHBK. dico, idem rectangulum 
e. nam quoniam parallelogrammum est HBEA, et 
4EB rectus est, etiam L AHB rectus est [I, 34]. 
liliter demonstrabimus, etiam angulos ad B, K, Z, 
sitos rectos esse. itaque ZHBK rectangulum est. 
demonstratum est, idem aequilaterum esse. ergo 

Vp. 17. %al snaziQa — 18. tarf] om. P. 17. xat] om. p. 
t] supra F. 18. H9] 9 e corr. p. 20. iaxi] ictiv P. 
HBEA] HJEA, sed J e corr. m. 1 F. AEB] B in 

. F. oQd^ — 22. AHB] mg. m. 1 P. 22. AHB] B in 
. F. 23. 9, Z, K F. 24. iaziv PB, comp. p. x6 

19K] P, F m. 1; om. Bp; xo ZH9K xsxQanXsvQOv V, P 
2. 




Euolidet, edd. Heiberg et Menge. 19 




290 ETOEiEISN a'. 

Tttgayavov aQtt iOzCv. xaX ztQiysypttXTat 'ai 
ABF^ xvnlov. 

Iltql rof 6o&ivra aga xvxlov rsr^ayavov irc^t- 
yiyQaazuL ' oJitp sSei icoirjeai. 



Elg 10 8o&iv zeTQayavov xuxAov iyyQaii^ 
"EeTco To So9lv TBTffdyojvov To ABFjd' Ssi drj aig 
to ABr^ TtTffayeivov xvxXov iyyQaipai. 

TtTfi/^e&a ixaTtga zciv AA, AB 8i%a xatic r« 

10 E, Z e^tjiista, xal dia jiiv tov E oaotiQa tmv AB, 
FA napdlXtjXog ^x^"^ ° E&, Siit S% low Z oxoziga 
Tmv AA, Br aapdXlrilos ^x^'" V ^^' ^aQaAkrjlo- 
yQttiifiov aQtt isrlv txastov rtav AK, KB, A®, &j4j 
AH, Hr, BH, Hz/, xal at antvavTiov avrmv nktv 

16 Qttl SriXovoTi Haai. [tieiv'}. xal iatl feij ietlv i} A/J 
T^ AB, xal ieri r^g fiiv AA rj^iOtta 15 AE, Trjg 
61 AB rjiiiesia ij AZ, fo^ UQa xal 7} A E r^ AZ' 
msre xal at anevavtiov tet\ aQU xal rj ZH r^ HE, 
inoias Si} Sti^ofiev, ozi xal ExuTtQa Trov H&, HK 

30 ixatEQa rmi' ZH, HE iOTiv fS?) ■ at TteeaQeg aga ttl 
HE, HZ, H@, HK teai «AAijAatg [fiWr]. o aQa 
xivtQa {itv ra H dtatfrij^Ari S^k ivi tmv E, Z, @, K 
xvxXoi yQa<p6(itvos rj^et. xal Sta tcov Aotjrrov arjfititav 
xal iipdiliiTai, rmv AB, BF, F^, AA tv&eiav Siu 

25 ro OQ&ag elvai Tccg JtQog rotg E, Z, ®, K yojviag 
ei yaQ «jm£ 6 xvxAos ra^ AB, BF, FA, /iA, ij 



A 



THl. BoetiuB p. 389, 6. 

1. iazlv\ comp. p; tai/ PBFV. B. r\\ m. 2 V. 
il ZK ^itftoi p. 13. KB] B mutnt, in E m. 2 F; BJt I 
14. BH, HJ] e corr. F. 16. t^aiv] Pj tiei BTp; om. 



t 



ELEMENTORUM LIB£R IV. 291 

ladrfttam est [I, def. 22]. et circum ABFJ circidam 
rGumscriptam est. 

Srgo circum datum circulum quadratum circum- 
riptom est; quod oportebat fieri. 

vni. 

In datum quadratum drculum inscribere. 

Sit datum quadratum ABrjdl. oportet igitur in 

BF^ quadratum circulum inscribere. 

secetur utraque Ajdlj AB in duas partes aequales 

Eij Z punctis, et per E utrique AB, Fz/ parallela 

icatur ES [I, 31 et 30], per Z autem utrique Ajdiy 

Fparallela ducatur ZK, itaque parallelogramma sunt 

AKy KB, A@y &J, AHy Hr, BH, 
HAy et latera eorum opposita inter se 
aequalia sunt [I, 34]. et quoniam 
AA=AB, et AE=^A^, AZ=^AB, 
erit AE = AZ, ergo etiam opposita. 
^ quare ZH = HE. similiter demon- 
rabimus, etiam esse H® = ZH, HK = HE, 
ique quattuor rectae HE, HZ, H0, HK inter se 
quales sunt. quare qui centro H radio autem qua- 
jet rectarum HE, HZ, H@, HK describitur circulus, 
iam per reliqua puncta ueniet. et rectas AB, BF, 
A, AA continget, quia recti sunt anguli ad E, Z, 
, K positi. nam si circulus rectas AB, BT, TA, 
^A secabit, recta ad diametrum circuli in termino 

;. ^B] £ in ras. F. 18. dnsvavtiov] P; dnevavtiov tWi F 
ed taat postea insert. comp.); anBvuvriov Hcai siaiv BVp. 




EH F, et 
(alt.) seq. 



rj aQa'] in ras. m. 2 seq. lacuna 3 litt. F. HR' 
corr. m. 2 ex HE. 20. ZH] HZ F. af 

6. 2 litt. F. 21. sta^v] om. P. 22. HE, HZ", H9, HK 
regorius. 24. JA\ mutat. in z/T m. 2 FV. 26. xiyi,v'(i B. 

19* 



STOIXEIflN S\ 






dia[iEtpa Toi) xvxkov apos opd^wg aic axffag 
ivrog nsettTat tou xvxXov oniQ aTonov iSt^x^- "ux 
«pa 6 xivTQCi rca i/ SiaCr^fiKTi Se ivl tmv E, Z, ®, 
K xvxilos ypftqDofi^vog TEfifF Tag ^B, BT, f/^, 
5 sv&f(as. i{pati>ETaL a^a avtav xal SSTai iyysyQai 
liivoq eis to ABF^ TBtqdyavov. 

Eig «pK To So^iv rtTQttymvov xvxkos iyyiyffaatai 
oXEQ iSei noi^^ftai. 



) IIcqI ro So&lv rsTQaycivov xvxiov iceqi 
ypdii-ai. 

"EGta ro 6o%^tv rttffdyiavov ro ABF^i' Set i 
kbqI t6 ABFA TETQdyiovov xvxkov xeQiyQd^ai, 
'Eni%av%9tii3ai yuq at AF, Bid TBHviTuiOav H 

r, XriXas xatd to E. 

Kttl incl tSTj iatlv fj ^A t^ AB, Kotvii Si 
Ar, Svo Sij ttt AA, AT Sval tats BA, AF tm 
slelv xaX ^dsis ^ AT ^daei rrj BF tet} * ytivla «pa 
imo AAV ymvitt t^ v%o BAF ter) iarlv r, UQtt vn 

D dAB yavia S^xti ritfiTiTttt ino t^s AF. b(to(cas 8 
SeC^oftBV, oti xttl Bxdart} rmv imoABr, BFA, T^. 
Sl%a TstfijjTat vno rmv AF, AB ev%Bmv. xaX ia, 
tat} iarlv i] iino jdAB yavta ry imo ABV, m 
iari t^s ft^iv vxo AAB fjfiiautt ^ vno EAB, d 



2. idcix»>i] PF; om- B7p. 3. jie^ipm p.i* P. H 

HZ, H9, HK ed. Basil. 4. Post K add. atnieiwv F 

rec. ttiiii] PF; itfH-et BVp, dA] AJ P. 6. ABT 
7, afB To So^h'] P; lo 30*^* &ftt Theon (BFVp). 9. I 

om. qi; 9' et litt. initialiB postea add. iaV, ut in seqnenfcibi^^ 
semper fere, 14. ^ncfitii^Ereai Vp; ^m£*«xft^<Joi ip' BA] 
JB P. 16. E] 9 P. 16. JA] AJ F. 18. tlaiv] PP; 

elei BVp. Dein mg. in V add. iKaxifix Ixai^. xai paais] 



ELEMENTORUM LIBER IV. 293 

Brpendioularis intra circulum cadet; quod demon- 
aratam est absurdum esse [lU, 16]. itaque circulus 
mtro H et radio qualibet rectarum HEy HZ, HSy 
IK descriptos rectas AB, BF, FJy JA non secabit. 
oare eas continget^ et in quadratum ABFjdi inscrip- 
18 erit. 

Ei^o in datum quadratum circulus inscriptus est; 
uod oportebat fieri. 

IX. 

Circum datum quadratum circulum circumscribere. 
Sit datum quadratum ABFd. oportet igitur cir- 
am ABF^ quadratum circulum circumscribere. 

ductae enim AF, BA inter se se- 
cent in E, et quoniam AA = AB, et 
AF communis est, duae rectae jdfAyAF 
duabus BAj AF aequales sunt; et 

^^ Ar=Br. 

»que L^^r= BAR ergo LAAB recta AT in 
uas partes aequales diuisus est. similiter demonstra- 
»imus, etiam angulos ABF, BFA, FAA rectis AFy 
iB m duas partes aequales diuisos esse. et quoniam 
,AAB=ABr, QiLEAB=\AAB, LEBA= iABF, 

xaxiga in ras. m. 2 F, supra scr. Ixar^^a §%ateQa m. 1 F. 
ctiv t^rj FV. 19. vno] (tert.) m.2 F. 20. JAB] B in ras. 
tt. 2 V. 21. JBT] P m. 1, F m. 2, V (F in ras. m. 2), p (r in 
ae.); AB, BT B, P m. 2, F m. 1. BTJ] P m. 1, F m. 2, 

r (B in ras. m. 2), p (B in ras.) ; B F, FJ B (punctis del. m. 2 ; 
ir in ras. m. 1); FJ P m. 2, F m. 1. FJA] T in ras. m. 
J V, r insert. Fp; FA P m. 1; JA P m. 2; rj.JA B; in B 
Dg. m. rec. y^. xa^* vno ABT, BTJ, FJA. 22. JB] FB 

p (non F). 24. ^ffrtv P. JAB] AJB F. rifiias^ag P, 

jorr. m. 1. EAB] litt. AB e corr. m. 2 V; AEB P; corr. m.2. 




294 STOIXEIfiN d'. 



di iiTth ABF rj(ii6Eia 17 wwo EBA, xal 17 vjtb EA, 
apa rjj vno EBA ieriv [Orj ' mOTB Kcel nlevQa 
EA xy EB iotiv fffij. ofioimg dij iJfi|o(i£v, ot( 
Exaztqa rav EA, EB [^ev&tiiav] ixrctiQa xmv Ef, 
6 E^ fetj iazCv. at xiecaQeq aga at EA, EB, EF, 
Ejd teai aXl-^lais etocv. o apor xEvrpo) ra E xeil 
SiaST^fiUTi ivl rinv A, B, r, .J xvxlos ypa<p6li£V0s 
^^ti xal dia Ttav loinmv atjiiBtiov xal iaxai TispiyE- 
yQa(t(iivos itep! 10 ABPA Tsxgaycovov. jtfptyEj^payOo 
10 tog 6 ABFJ. 

TltQl To do9'^i' «pa rsTQayiavov kvkXos «epi; 
yQaittai ' oittQ idei noiijaai. 



3 



1 



'Icoaxslig xgtyiavov ev3xijaae9'ai Ixov i: 
15 tipav xmv «poff xfj ^aest ymvimv StnKaeio' 
xijs KoiJt^g. 

'ExxiiaS^a xig evd^sCa 17 AB, xal T*TfiiJe#o xttra 
xo r aijfittov, Saxs ro vno rmv AB, BT wptEjjo- 
ft,£vov 6g9'ayc6viov taov elvai Tija «wo riig FA rfTpa- 
20 ymvip' xai xivT^m xa A xal diaSxijitaTi ta AB 
«Aog ytyQtt^9(o b B/dE, xai ivrj(f(i60&N tCg xov B^i 
xvxXov xjj AF sv&iCa (t-ii (tei^ovi ovari x^s xov B^i. 
xvxlov SiafiixQov Jffi) sv&sta tj BA' xai ixs^evx^t^Ot 



X. Proclm p, S04, 1 



1, Vfcst*] e corr. m. 2 P. E^B] E^^Jl' F. 8. S^ 

om. p. maic xal nlEi>eci] Mui Bp, 3. EJ] A in ran. HC 
V; A% F; £B ana Bp. Post G^ iu V add. itlfupa; idei 
F m. -2. BB] B in rtta. m. 2 V; E-i Bp. 4 E,rf,'EB] P, 
F m. S, V in raa. m. 2; ET, E/J B. F m. 1, p. ei!*«iDv] 

om. P. Er, EJ] P, F m. 2, V iu ras. m. 2; E^, EB B, 



ELEBfENTORUM LIBER IV. 295 

i L EAB — EBA. quare etiam EA «» EB [I, 6]. 
liliter demonstrabimus, esse etiam EA ^ EA, 

EB — Er.^) 
que quattaor rectae EA, EB, EFy EJ inter se 
|ual68 sunt. quare qui centro E et radio qualibet 
itamm EA, EB, EF, EA describitur circulus, etiam 
r reliqua puncta ueniet^ et circum quadratum ABFA 
oumBcriptus erit circumscribatur ut ABFA. 
'Exfp circum datum quadratum circulus circum- 
iptus est; quod oportebat fieri. 

X. 

Triangnlum aequicrurium construere utrumque 
gulum ad basim positum duplo maiorem habentem 
iquo. 

Ponatur recta aliqua AB, ei m 

puncto r ita secetur, ut sit 

ABxBr^TA^ [11,11]. 
et centro A radio autem AB cir- 
culus describatur BAEy et in 
BAE circulum aptetur recta BA 
rectae ^jTaequalis, quae diametro 
circuli^^^Jmaior non est [prop.I]; 

1) Uidetur enim scribendum esse EJ, EF pro EF, EJ 
. 4. 

m. 1, p. 6. tarj — EB] om. B, in ras. insert. p. 7. 

d, ££, £r, EJ Gregorius. Post J mg. add. CTifAsioav F. 
«€9iy£ypa(p^(o (og 6 ABFJ] om. Bp. 11. yiyoaTnai p. 
. AB, BF] F; alterum B om. B, m ras. m. 2 V; prius B 
d. m. 2 Pp. 20. %6vtQa} filv tm A SicccvTifiaTi 6i V. 

. AT] r in ras. m. 2 V. ' svd-sia] om. p; m. 2 B. BJE] 

supra m. 1 P; JBE Bp, V {JB in ras. m. 2); BJE F. 




XTOIKEliiN S . 



> ArJ^i^^ 



vov xtJxAog 6 AFA. 

Kal iatl xh imo riav AB, BF leov istl %a axh 
zfts AF, f«j) 6s ^ Ar vii B^, To aga vno zmv AB, 
5 BT taov iazl ta aitb r^g BA. xal iTcel xvxXov zov 
AT^ efAijJtrai' u iXfjfisiov ixtog t6 B, hbI uno zov 
B nQos zov ATA xvxkav «poiJjrsntroxaHi dvo ev&siat 
ctC BA, BA, xal tj fiiv «vtwv rifivei, ^ di ^rpoff- 
itinxit, xai icTi ro vno zav AB, BF Heov za nreo 

10 %^e B^, ij BA KQa ifpaarszai, tov Arjd xvxAotr. 
ixel ovv ifpdnrsrai. (liv tj B^, dsto Sl t^g xara zo 
jd iaa<p^g Si^xrai tj AT, 17 ap« vno BAF yavia itsri 
ierl r^ iv r^ ivaXlai xov xvxlov rfi^^fiari ycavia TJj 
imo jJAF. ixtl ovv Tot; iazXv ij vao BAF ztj ioto 

15 AAT, xoivij 7tQO0XE£0&ca fi imo T^A' oA*j opa ^ 
vno BAA [atj ierl Svtsl ratg imo TAA, AAT. a/lA« 
rafg vjto TAA, AAT Heri idrlv ^ ixTog 1) vn:b BTA- 
xal 7} iao BAA apa larj iorl r^ vnb BTA. dXla 
{] iinb BAA tjj vnb TBA ietiv fojj, inBl xal nXsvpa 

20 ij AiJ TJ} AB iSTiv tor]' roffTE xal tj vnb ^BA rij 
vnb BTjJ iattv ieij. aC rQttg aga al vno B^A, 
^BA, BTA tSat, dXXriXais eioiv. xal ixEl fffij iarlv 
ij vnb ABT yavia rfl vno BTA, teti ietl xal nXtvpu 
rj BA nXtvifa zjj ^T. dXXa ^ 3A zy TA 1 



1. A^ m raa. m. 2 V. ^F] FJ P. ATA] TA in 

laa. m. 1 B, ut etiam supra quaedam. 3. ABT PB Fp, in 
PFp tn. 1 inaert. B. 4. t^s AT ~ 5. tcd bti*] bia P, aed 

corr. 4, Post pritia AT \a ¥ add. □ m. 2 et in mg. xfZDu- 
yuvcB m. 1. B-J] JB F. AB, sr] Pp, prina B m. 2 in 
raB.'V; ABr B. cort. m, 2; F, corj. m. 1. 6. to B] corr. 
ei T^ B aeq, ras. S litt. V. T. atjoanenzioKaaiv B. 8- BA] 
P; BFA Bp, V {.^ 111 raa. m. H), F (rj in raa, intercedente 
rae, 1 litt.). 9. ieziv P. tm*] om. P. AS, BT] alt B 



£L£BI£NTORUM LIBER IV. 297 

Incantar A^^ dFj et drcum AFd triangolum 
umscribatQr circulus AFA [prop. V]. 
et quoniam AB X BF — AI^, et AF ^BAjeni 
X BF — BAl^. et quoniam extra circulum AFA 
iptnm est punctum quoddam By et a £ ad circulum 
'd adcidunt duae rectae BA^ BAy et altera earum 
it, altera adcidit tantum, et ABxBF^^^BA^ 
A Bd contingit circulum AFA [III, 37]. iam 
niam Bd contingit, et a ^ puncto contactus pro- 
ta est dFy erit LBAF^^^ AAF, qui in altemo 
Qiento positus est [111, 32]. iam quoniam 

LBJr—AAr, 
imunis adiiciatur L FAA, itaque 

L BAA = FAA + AAR 
FdA + dAr^BFA extrinsecus posito [I, 32]. 
re etiam L BAA — BFA. uerum 

L BAA = FBA, 
& AA = AB [I, 5]. quare etiam L ABA = BTA. 
ine tres anguli BAA, ABA, BFA inter se aequales 
it. et quoniam L ABF = BFA, erit etiam 

BA = Ar [I, 6]. 



as. m. 2 V; ABTFB (corr. m. 2), Pp (corr. m. 1\ 10. 

] ^ 6 corr. F. V ^^] supra m. rec. i^. 11. iml ovv] 
inei P. fiivl PF (tov xvxilov 7^ BJ bv^biu xaTa to ^^ 
F); om.V; tov xvxilow Bp. 12. aq^rjg Theon (BFVp). 

IfFTiTi^ P. Tj Iv] m. 2 V. 14. BJr] P, V m. 1; FJB 
Vm.2, F in ras. 15. JAT] F in ras. m.2V. 16. BJA] 
in ras. m. 1 B. iaviv P. 16. dAT] JAH ip (non F). 

7. iatlv rj] in ras. m. 1 p. intog] om. p. 18. %al ^] 

ifu P. BJA] AJB P. apal om. P, m. rec. F. 

(9 tmi F. icxCv PB. aXX' FV. 19. F^^] V m. 1; 

zl Vm. 2. icri iaxCv BFp. 20. iVi? iaxiv p. .dB-/!] 

Ii< P, F m. 1 (corr. m. 2). 22. bIcIv] PF; bM BVp. 

Itts^i' V, sed V eras. 24. ^rZcvpa] om. p., m. 2 B. aXk' F. 



298 rroiXEiHN *'. ^ 

i^ffij * xal ^ ZV apK T^ r^ ieziv fff?j ■ mStB xal yav^K 
■}} vno F/iA yavia Tij vno jJAF iexiv Hati' ai kqr 
imo rdA, ^ AT zijg v%o AAT elei SiJtkaaCov?. 
tof] di 7} vao BTA rais vao TAA, AAT' xaX 
5 ^ vno BTA «pa r^g vno TAnJ iari 8ial^. teti 
Sl 7/ vxo BT^ sxaTif/a tmv mch BAA, jdBA' xal 
ixariffa apa rmv V7tb BAA, ^BA rijs vk6 jdAB 
iexi diTcXij. 

'ISoSxEles oiQtt XQiyavov ewiaraTat rh ABA i%ov 
10 ixatd^av ziBv jc^og zfi AB ^dSEt yavi.(Sv SmXaeiova 
t^g Xoia^g' oxsg ^Sei ironjffat. 



i 



Eis thv So&ivra xvxXov HBVzdycovov ie6- 
iiXsvQov zf xal iaoydviov iyyQdiiiKi. 

IB "EffTo 6 So^eis avxXas 6 ABTAE' dst dr} eig xov- 
ABTAE xvxXov ntvtdyavov CeonXBvqov te xaX lei 
ytoviov iyypd^ai, 

'ExxcCa&a tpCycovov iaoaxsXig ro ZH& 8i%Xaeiovtt 
Ixov ixaziifav zmv npog Tofs H, & yatvimv Tijs JC<fOs 

20 T^ Z, xal iyyeyQatp&m dg zov ABTAE xvxXov Tp 
ZH® TQtyavm ieoymviov XQiyavov th ATA, maxt 
TJj [i^v Jipoff xa Z yavCa fffiji' elvai zijv vah TAtiJ, 
ixariQav Sh xmv irpog TOfs H, & fffjjv ixaxiqa xmv^ 

XI. BoetiTia p. 389, 10. 

1. TA] Ptf, V in ras. m. 2; AF Bp. S. yc^Ca] om. V. 
3. Jjr] (alt.) P, F {BQpram. S: r/JA), V in raa. m. 2; FA^ 
Bp, Binlaaioi F. 4. Se] Si %aC V. ^] aopra m. 2 P. 
r^AI Pcp; in ras. m. 2 V; FAJ Bp. i3Ar'\ FJA Bp. 
■ff^] Smlij aifa Bp. 5. Sqb] om. Bp. PAJ] ia — " " 

r e corr. F. l«w PB, eomp. p, SisX^] om. B 

Ktti'] om, P, 7. ^AB} BAJ P. 9. ooWaraioi V. 



1 

^v J 



ELEBCEKTORUH LIBER lY. 299 

0appo8iiima8y esse jB^ — Pji. itaque etiam 

rA — r^] 

eidam L F/IA — dAF [I, 5]. itaque 

r/iA + dAr — 2 /lAn 

V/l — FjdA + ^^r. itaque etiam 

BrA~2rAA. 
r^^ Bj^A — ^BA. ergo uterque BAA, 
duplo maior eet angulo AAB. 
;o triaiigalus aequicrurius constructus e&t ABA 
jm angolom ad AB basim positum duplo ma- 
liabens reliquo; quod oportebat fieri. 

XI. 

datam circulum quinquangulum aequilaterum 

oiangulum inscribere. 
datus circulus ABF/IE. oportet igitur in cir- 
ABFAE quinquangulum aequilaterum et ae- 

pilum inscribere. 

4^ f^ construatur triangulus aequicru- 

Ariu^ ZHS utrumque angulum ad 
H, S positum duplo maiorem ha- 
bens angulo ad Z posito [prop. 

^X], et in circulum ABFAE tri- 

) ZH0 aequiangulus inscribatur triangulus 

ita ut sit L VAA angulo ad Z posito aequalis, 

le autem AFAj FAA utrique angulorum ad 




V m. 2; AJB P. 10. BJ p. 16. icxm — 17. ly- 

] om. JP. 19. Ixarepay] om F. nqoq xot^ H, 

wv] lommv P. 20. tw] (prius) to B, F m. 1 (corr. 

22. ra] x6 B. 23. (xaripav] B%axioa {am ras.) p, 
^ P. ' xmv] in ras. p; xriv B. ^%ttxi^ct] sntcxiQtxv r 
)rr. p. xmv] qp, aga xav F. 



i 



300 ZTOIXEIHS 3'. W 

iiab AF^f F/iA' xal iieatdpu apa tmv vao AFjd, 
r^A T^s wsro TA^ iart SinXtj. rfrfi^ff&ra dij ixK- 
ttffd T&v V710 AF^, FjdA di^a vnb exKiiffaq Trov 
FE, jJB £v%Etmv, xal instsvx&oiaav «f AB, BF, 
5 [r^], ^E, EA. 

'Endl ovv ExaTsga xav vab AV^, F^A ycovtav 
SixAaaiav iatl i^g vnb FA^, xal tsTfirjfiivat tl«l 
dCxa vjto tcov TE, /!B sv&cimv, aC nivts apa ytD- 
vCai at vjtb ^AT, AFE, EF^, F^B, BJA Ceai ai.- 

10 Xrj^aig eieCv. at S\ teai ymviat hd Catov aeifi^epticav 
^fjSijxafftv at nivTE ap« nEQiipipEiat at AB, BF, 
FA, ^E, EA COat aXXriXais tiaCv. vnb di tas taas 
ntQtipiQEias Caat sxi&itat vnoriivovotv at nivts a^a 
iv&tlat ttt AB, BF, F^, ^ E, EA Caat aXk-qXats 

15 sCaiv CaonXsvQOv a^a iarl t6 ABFjdE ntvtayiavov. 
Xiya iJ^, ort xul looydvtov. intl y&Q ^ AB arspt- 
(pigtitt tfi ^E negiqieptia ietlv tatj, xon/^ ngoaxtia&a 
ij BT^' oXrj aQtt rj ABF/d ni^Kpiptia oAij t^ E^VB 
ntQi<ptpfia ietlv Cari. xal pi^ijKEv inl fiiv tijs ABFA 

20 ntQi<ptQtitts yaivia rj vnb AE^, inl Si tijg E/irB 
nsQtipeptias y<avia i) vnb BAE' xal tj vno BAE 
apa yavitt tfj vnb AEid iotiv tejj. Sta ta avta 
Sij xttl ixdatTj tcov vxb ABF, BVA, FAE yavimv 
ixatiga zav vnb BAE, AEjJ iativ taij' iaoyeiviov 

25 «p« ietl tb ABr/iE nsvtdymvov. iSsix&ti Si xal 
iaonXsvffOv, 

l. POBt rjA mg. m.2 add. ymvi6v F. 2. c^g v9io VAA] 
om. p. S^] om. Bp. 3. ixaiifag] mg. m. 2 V. 4. r£j 
E e corr. P. dB] z/E F; corr. m. rec. B. r^j] om. V. 
7. loTly P. tleCv P. 9. Erj] J in ras. m. 2 P. r.JB] 
in rafl, F; r in raa. m. 3 P. B^J] m raa. F, e corr. m. 2 

V. oll^lffis ilaiv] dXljj in tas. F, leliqua abaumpta ob pei^- 



^ 



i 



ELEBfENTORUM LEBEB IV. 301 

positorcim aequalis [prop. II]. quare etiam 

L AFJ — r^lA — 2 FAd. 

AF^y FAA rectis FE, ^B in binas partes 

18 secentor [1, 9], et ducantur ABj BFy AE, 

iam qnoniam anguli AFA, FAA duplo maiores 

Qgnlo FAA et rectis FE, AB in binas partes 

fis secti sunt, erit AAr = ATE = ErA 

B — BAA. et anguli aequales in aequalibus 

8 oonsistunt [m, 26]. itaque quinque arcus 

ir^ FA^ AE, EA inter se aequales sunt. et 

equalibus arcubus aequales rectae subtendunt 

)]. itaque quinque rectae AB, BP, FA, AE^ 

ater se aequales sunt. itaque quinquangulum 

^E aequilaterum est dico, idem aequiangulum 

nam quoniam arc. AB = AE, communis ad- 

r arc BFA. itaque arc. ABFA — EAFB. 

arcu ABFA angulus AEA consistit^ in EAFB 

L BAE. quare etiam LBAE = AEA]jn, 27]. 

de causa etiam singuli anguli ABF^ BFA, 

utrique angulo BAE, AEA aequales sunt. 

aequiangulum est quinquangulum ABFAE. sed 

istratum est, idem aequilaterum esse. 

Lin. 5 nidetar delendum esse FJ cum Gregorio. 



aptum. 10. de] d' BV. 12. dciv] ictiv V. 16. tao- 
] litt. lao' in ras. m.2 V. 17. xy JE nsgivpSQsia] om. F, 
m. 2: T^ EJ nsQiq^SQsia. tari iaUv V. 19. tari iaxi 
20. EJTB] BTJE F. ' 21. rj vno BAE] mff. m.2 F. 
comp. supra scr. m. 2 F. 22. yoivCa uQa Y. tarj 

23. xat] om. BV. 26. iauv PF. 



302 ETOIXElflN S'. 

Elg apa xov So&ivta Hvitkov itsvtdymvov leo- 
jcksvQov TE xttl ieoywviov eyysytfaxtai' ojtcQ iSet 

Ttoiijeai. 

'!>'■ 

6 IIiqI thv do&Evra xvxXov nevTdyavov leo- 
^Xevqov te xal iaoywvi.ov ZsgtyQailiai. 

"Earia 6 do&tls kux<Ios 6 ABP^E' Sei di «c(fl 
zbv ASr^E xvxKov ^EVtdyavov CeonXivpov te xal 
leoymvtov nEpiyQail/ai. 

)0 NEvo^e9Gs rov iyyEyqa^^ivov XEvrayiavov xav 
yaviav erjfiEtcc i;« A, B, P, ^, E, aete i'0ag stvai. 
tag yiB, Br, rj, JB, EA nEQi^EQtiag- xal Sta 
tAv A, B, r, /J, E Tix&G}0av lot xvxXov itpaxrofisvttt 
al H&, ®K, KA, AM, MH, xal Ei^tp&o tow ABF^E 

15 xvxXov xivTQov To Z, xal inE^BVx&aeav ai ZB, ZK, 
Zr, ZA, Z^. 

Kal iitBl ii (isv KA Ev&stcc ifpdjitErai rou ASr.dE 
xata ro F, «;ro Si rov Z xivrQOV inX zijv xara 
r inatpiiv hti^Evxtat vj ZF, ^ ZF «pa xd%Er6g it 

20 inl ziiv KA ■ opQ^rj apa iarlv ixarifia tcov sipog 
r ycnvi^v. Sia td avTa tf^ xal at ZQog totg B, Ji 
etjfifioig ytoviat oQ&ai eieiv. xal iTttl opfrij ieriv rj 
vxis ZFK ycavia, rb &Qa dnb r^g ZK taov ierl Torg arao 
rmv Zr, FK. Sta rd amd Svj xal tois ano tav 

26 ZB, BK [aov ierl ro dab r^g ZK' Sgte to: dnb rtav 



^^E 
d i JB 

? -tSM 



SU, Boetins p 



1. mJKlo*] corr. ex jhjkIos m. 2 F. 2, -ce] om. V. 
jrot^OQt] Scitet V; yp. aeiiui mg. m. 2 F. 7. ^BTJE] 

m ras. m. 2 V. 8, ABFJE] E in ras. m. 2 V. 11. ot/- 

H([a] -n io ras. m. 2 V. 13. AB, TJ, JE P. 14. MH] 
MN F; oorr. m. 2. 15. ZB] B e corr. m. 2 F. ZK] ZH 



I 



ELEHENTOBUU LIBEB IV. 303 

rgo ia datam circolam quinqaangalum aeqoi- 
1 et aeqoiaBgalum iascriptam est; qaod opor- 
fieri. 

xn. 

ircam datam circolum qmnquangulum aequilate- 
Bt aequiangulum oircumscribere. 
t datus circuluB ABFdE. oportet igitur circum 
!^£ circulum quinquaDgulum aequilaterum et 
angulum circumscribere. 

igamus, puncta angulorum quinquanguli inscripti 
• XI] esse Af B, F^ A, Ey ita ut arcus AB^ BFy 
^Ey EA inter se aequales sint; et per A^ B, 
f; E circulum contingentes ducantur H0, 0K, 
AM, Afir[in, 17], et sumatur circuli^Br^fi 
im Z [111,1], et ducantur ZB, ZK,Zr, ZAy ZA. 
i quoniam recta KA circulum ABFAE contingit 
, et a Z centro ad F punctum contactus ZF 
21 ducta est, ZjT ad KA perpendicularis 

jg est [III, 18]. itaque uterque angulus 
^M ad r positus rectus est. eadem de 
/^ causa etiam anguli ad £, ^ puncta 
positi recti sunt. et quoniam L ZFK 
•l' ^ rectus est, erit 

ZK^ = Zn + rK^ [I, 47]. 
a de causa etiam ZK^ = ZB^ -{- BK^. quare 

Zr] r in ras. F. ZA] ZA 9. 17. i|l sl 9, supra 
l. Post ABFJE add. nvnXov V, supra F(comp.), P. 
v] rmv comp. V. Post KA in F add. m. 2: sv^siav, 
PF; om. BVp. 21. xaO m. 2 V. 23. ZFK] K 

ante Z ras. 1 litt. V. t^s] om. Bp. 24. x&v] T^g 

V. Zr, FK] r prius et K m. 2^V. 26. tcovlaxC] 
iexCv F. ZK tcov V. wgxb xd] PF; xaaQu 

xmv] om. Bp; r^j V. 




asflP 



304 LTODCEIiiN d'. 

Zr, rK-Totg am rav ZB, BK itSxtv fffa,"m^? 
ajro r^S ZF rto asio r^s ZB ^driv faov Aomot' 
affCL ro «Ko r^i; TK Ta ajto T^g BK iOTiv fffov. fffjj 
ftpB ^ BKTfj FK. xal ijtel /"07) iGtIv 17 ZB r^ Zr, 
B xal xoi-v^ V ZJf, Svo dij ul BZ, Zfi^ Sval Tatg fZ, 
Zff fffat cMv xkI fiaais V ^^ /3aff£i T17 fiT [^ffriv] 
fffj;- yra!'i'a ap« ^ ftei/ irno BZK [yavia] tfj v«6 
KZr iOTLv ia^- ii St vwo BKZ r^ vita ZKP- 
SiitX-^ a^a Tj (ilv vitu BZT Trjs vao KZP^ tj 8h 

10 BKF Ttjs uTo ZKF. Sia va avxa Srj xal ■^ 
ino rZj4 T^s v%o rZA iaTi SiTtXij, ^ Sl vao A 

• TTjs vao ZAF. xal ixel latj ietlv rj BV XEQttpigsi 
r^ rj, fai) ^ffri xal ycovia r} vao BZFt^ vjto rZJ. 
xtti iattv 7] fiiv vTth BZr T^s vnb KZF SiTtlij. 

16 Se vxo JZT r^s uKo AZT' tati aqa xal 
KZT t^ vno AZT- iatl Sh xal ^ vno ZTK j-w 
tfi vno ZTA fffij. tfuo ^17 tQiytova iati ta ZKi 
ZAT Tas Svo yaviag tatg Svel yiaviai.s taag ixovra 
xal fiiav itXsvQUV (iia nliVQa l^Oijv xoiv^v avrwv 

30 T^v ZT' xal rcg Xotitas a(/a alevQtts tttig Xotnats 
ni.£vQttts faag e^it xal tjiv Xotni}v yavCav rjj koimi 
ytovCtt' (ari apa ^ (liv KT Bv&sta ty TA, ^ Si vno 
ZKT yuvCa Tjj vno ZAT. xal inal taij iaxlv 



.rne rK] m I 
r in ras. F; r^s KT B. Aiite 11» in F add. m. 3: li 
BK] B in ras. F, teov iatlv V. 4. BK] FK P. 
BK P, 6, SvBi] Sva P; Svafv V. 6. ti<rl BVp. 

iinte r ras, 1 litt., Jt m. 2 V; KF P. Itsnv] om. P. 

(ii*] m. 2 V, BZK] P; BKZ Bp et FV (eed KZ u 

ymvia] om. P, 8. KZP] e corr. P m. 2; FKZ Bp; ZS 
in raa.FV. BJtZ] P; BZX Bp et e corr, FV. """' 

P; rZK Bp, e corr. FV. 9, -f ZT] K bx raa. 






ELEBiENTORUM LIBER IV. 305 

Zn + rK* — ZB^ + BK\ 
am Zn — ZB\ itaque FK^ — Bli:». itaque 

BK~rK. 
uomain ZB — ZP, et ZK communis est, duae 
yb BZ, ZK duabus TZ, ZK aequales sunt; et 
— FK. itaque L BZK — KZF [I, 8]; et 

LBKZ^ZKF [I, 32]. 
le Z-ilZr— 2 XZr, t JJJfr=2 ZJCr. eadem 
ausa etiam L TZA — 2 FZ^, Z. ^AT—^ 2 Z^F. 
uoniam arc. Br=rj, erit etiam 

LBzr^rz^ [III, 27]. 

HZr— 2Ji:zr, Z.^Zr— 2 ^Zr. itaque 

Z.Jfzr=^zr. 

m etiam Z. ZPX = ZFA. itaque duo trianguli 
r, ZAr duos angulos duobus anguUs aequales 
mt, et unum latus uni lateri aequale, quod utri- 
16 commune est ZP; itaque etiam reliqua latera 
[uis lateribus aequalia habebunt et reliquum an- 
im reliquo angulo [I, 26]. itaque 

Kr=rA, LZKr=zAr. 



kS. m. 2 V. 10. BKF tfjs] litt. KT trjg in ras. m. 1 B. 
rZji] A in ras. m. 2 P. dAF] in ras. m. 2 V; u4 in 

m. 2 P. 12. ZAF] in ras. m. 2 V. 13. Post FJ in 
,S add. nsQKpsQS^oc. iaxlv P. BZF] in ras. ip. 

5Zr] in ras. F; BZT dtwZiJ p. dinX^ om. p. 15. 

"•] in ras. V; FZJ SinXij Bp; dmXrj in P add. m. 2. 
^J AZ in ras. m. 1 p. 16. KZF] KZ in ras. P; KZF 
u BFp, V m. 2. Tj] T^s P. AZr] .i et F in ras. 

V. iarl 61 — 17. tari] P; om. Theon (BFVp). 17. 
1] -4 in ras. P. iazi] om. P. 18. ZAT] FZA P; 

i F. Svai] SvaCv V, 8vo B. Post 1)^0»'*« bab. V: 

iQuv ixaxiQa, idem F mg. m. 1. 19. fii^ nXsvQ^] snpra 
F. 22. FA] Ar F. 23. ywv^a] om. p. Post ZAF 
1 litt. V, yoaviloc supra scr. m. 2 F. 

iaolides, edd. Heiberg et Menge. 20 



306 STOIXEIfiN S'. f 

KF tfl FA, Sinkfj uQa ij Kyl T^g KV. dia r« avra 
dtj dsix&^6ETai xtti 7] &K Ttjg BK dtTtXij. xccl iativ 
71 BK t^ KF Herj- xal tj &K uQa ty KA ietiv iaij. 
bfioCas S^ Sux&^ettat xal txacrij ttov &H, HM, 
■T MA ixatBQa rmv &K, KA ttsri ' leoaXevQov aQcc iatl 
To H&KAM ntvtaymvov. Xiya tf^, ott xal iooyaviov. 
iarel yuQ terj iatlv 4 vjtii ZKT ymvia tjj vnb ZAF, 
xal iScix^ ^^S f^i^ v3to ZKT Siitlil ij vTto &KA, 
T^S Si vjto ZAF Siitkii ii vao KAM, xal ij vjtb 

10 @KA UQa zij VTtoKAM ietiv fffTj, biioicog Si] rfEij- 
a-ij8irat xal ixdettj tmv vno K&H, &HM, HMA 
ixaziQa tmv vno &KA, KAM fijTj- aC nivtf. apa 
ymvCal aC v%o H&K, &KA, KAM, AMH, MH® 
ieat tUAijiats titsCv. ieoymviov aQU iszl tb H&KAM 

15 ntvtdymvov. iSsCx%r) S% xal (eonXevQov, xal 
yiyQaxtai mgl tbv ABVAE xvxKov. 

[JJfpi rof SoQ-ivta uQa xvxKov itavrdymvov liH 
TtKevQov te xal Ceoytavtov MQiyiyQantai] ■ ohbq li 
itoirjeai. 

ao ty'. 

Eig rb So&hv aevrdymvov, o iattv ieoicXe 
q6v n xal iGoymviov, xvxXov iyyQatiiat. 

"Eerm To tfo^iv JTEVTayojvov laoTcXevQov re > 
iaoydvtov xb ABPAE' Sei Sij eig ib ABF^ETtevrA 

25 ymvov xvxkov iyyQaii^at. 



Xm. Proclns p. 173,11. 

1. KF} (priua) rk F. 2. tffij^^oETai] notat. punctifl 
jian om. p. Ante SixXn m. 2 ftdd. Caxiv F. laxip} P 

JnJ IBilx^v f<"i Theon (BFVp). 3, Cijij] P; %<,{ iezi diiA^ 
71 ^iv KA t^i KF ri S\ 0K t^s BK TheoD (BFVp), tb] 

T^« comp. p. 4. Ante %ai iu F add. oti m. i. BH] P; 



ELEMENTORUM LIBER lY. 307 

:oiiiani Kr^FAy erit KA = 2 KF, eadem ra- 
demonBtrabimos, esse etiam SK = 2 BK. et 
* Kr. quare etiam 0K ~ KA, similiter demon- 
[mu8| esse etiam singolas rectas BHy HM, MA 
le BK^ KA aeqnales. itaque quinquangulum 
r^Af aequilaterum est. dico, idem aequiangulum 
nam quoniam L ZKr= ZAFy et demonstratum 
esse L^KA^2 ZKr, et KAM=2 ZAF, 
tiam L BKA^KAM, similiter demonstrabimus^ 
singulos angulos KBH, BHM, HMA utrique 
o BKA^ KAM aequales esse. itaque quinque 
i HBK, BKA, KAM, AMH, MHB inter se 
des sunt itaque aequiangulum est quinquangulum 
lAM. sed demonstratum est, idem aequilaterum 
et circum circulum ^£Fz^£circumscriptum est. 
rgo circum datum circulum quinquangulum ae- 
lierum et aequiaugulum circumscriptum est; quod 
ebat fieri. 

XIII. 

i datum quinquangulum^ quod aequilaterum et 
angulum est, circulum inscribere. 
it datum quinquangulum aequilaterum et aequi- 
um ABFAE, oportet igitur in quinquangulum 
^AE circulum inscribere. 



'; HS BVp. 5. M.il M in ras. m. 2 V. Ante Hcri 
r m. 2: hziv. hti] iaxiv P. 9. jj] (prius) om. p. 
\a\ iariv, supra scr. aga m. 2 F. ry] tjjs Bp. ictiv] 
11. Ante xa^ F m. 2 ins. oti. KSH'\ e corr. F; 
H in ras. m. 2 V; SKA P. 12. Ante Ccrj msert. iativ 
2. 16. TtfifiysYQaTttai] om. Bp. 17. nsgi — 18. nBifi- 
Mtai] om. codd.; add. Augnstns. 23. Post nsvtdytovov 
) iativ BVp, F m. 2. 24. sCg to] seq. ras. 1 litt. P. 

20* 



308 ETOIXEIiiN 6'. 

TeTfi^e&ii} 7«p ixatEQd tmv vxa BF^, r/iSyiO' 
vimv dixa vxo ixatd^as tiav FZ, ^Z ev&Hmv xal 
«jro row Z arjfitCov, xa9' o avfi^dlXov<ftv ali.ijlais 
ai rZ, ^Z evd^etai, ijtetevx&maKv at ZB, ZA, ZE 
5 ev&htdi. xal inel tanj iatlv rj BF tfj F^, xoivij dl 
7] rz, dvo Sri al BF, TZ Sval- tafg ^T, FZ taai 
elaiv xal ycavia ij vna BFZ yavia ttJ vxo ATZ 
\iaxi.v\ tat}' ^uais «pa ri BZ ^dati t^ ^Ziativ fo% 
aal to BrZ T^iyovov ta ^d VZ tffiyavs} ietiv teov, 

10 xal al Xoi-xat yaviat tatg Xomatg yatviaig tOai iOov- 
tai, itp' ag at taai nlsvifal vnozfivoveiv teri «pa ij 
VTCo TBZ yavCa ty vito T^Z. xal iitti SiitXil ietiv 
r\ vno r^E t^g vitb F/JZ, tar} di rj fiip VTto FidE 
zy vitb ABF, ti 8% vnb FJZ tij vab FBZ, xal rj 

IB vjtb FBA oLQa trjs vab TBZ iati Sml^' tatj a^a ^ 
vm ABZ yavia tg uah ZBT' r aqa vnb ABT 
yavia Siya tiTjirjttti. vnb r^g BZ sv&eiag. Ofioias 
dij dfiji^&ijffftai, ott xKr exazi(fa tav vnh BAE, AEA 
Sixa titfiTjztti vxh exazigas zmv ZA, ZE ev&stmv. 

20 ^);3-Mffav Sij «Tcb zov Z e-tjfieiov inl T«g AB, BT, 
TJ, JE, EA iv»eias xud-szoi. at ZH, Z&, ZK, 
ZA, ZM. xal iitel tat} iazlv rj vno &TZ ymvia trj 
^nb KTZ, ietl Sh xal opftij ij iixb Z&T [op&jj] tf/ 
vnb ZKT terj, Svo Si) t^iycavd iati ta Z&T, ZKT 

25 T«s Svo yaviag Svel yavitttg taag ^j;ovra xal fiiav 
TiXevQav (ua xIbvqu tatjv xotviiv airtav Xfjv ZT v«o- 



2. vrtoj om. q>. dZ] ZJ Bp, Yinrae. in.2. 6. fottt — 8. 
tvn (priQB)] mg. m, 1 F. 7. iliilv\ P; tl<s( BPVp. 8. iciiv fin]] 
P in teita m. I, Bp; tofi lerl V, F mg.; Toji P, aZ\ J« 

P, corr. m. rec. 9. BFZ] in rag. V. JTZ] JZrP. 

taov iaxC Y. 13. rSZ] ETZ p; TBZ F m. 1, ABZ gj, corr. 
ra. roc, Siii}.ii\ om. V. IS, TJZ Sinlri seq, ras, 3 litt 



J 




BLEUENTORVH UBER IV. 309 

lecetTir enim aterque angnluB BFJ.rJE in bmaa 
ee MqoaleB ataique recta FZ, ^Z, et i Z puncto, 
ao rectae TZ, AZ inter se concurrnnt, dacantur 
le ZS, ZA, ZE. et quonism Br—F^, et rZ 
mmiia est, duoe rectae BF, FZ duabus jJF, TZ 
lalea ennt; et L BTZ — ATZ. itaque BZ = ^Z 
[I, 4], et A BTZ — ^rz [id.], et 
reliqui anguli teliqois angnlis ae- 
qaalea erunt, aab quibas aequalia 
y^ latera aubtendunt [id.]. itaque 

/. FBz =- r^z. 

et quoniara LT.dE — ^rAZ, et 
l TAE = ABT, l TAZ — TBZ, 
laam LTBA — 2 TBZ. itaque L^BZ-= ZSr.') 
ae L ABT recta BZ m duas partea aequales 
EniB eat siniiliter demonstrabimus, etiam utrumque 
nlom BAE, AEA ntraque recta ZA, ZEm binas 
■jes aequales diuisum esse. ducantur igitnr a Z 
cto ad rectas AB, BT, TJ, AE, EA perpendicu- 
B ZH, Z9, ZK, ZA, ZM. et quoniam 

L&rZ~ KTZ, 
.Z&r= ZKT, quia recti sunt, duo triauguli Z©r, 
'.T duos anguloB duobus angulis aequales habent 
inum latus uni lateri aequale, quod utriusque com- 
ue est ZT sub altero aequalium augulorum sub- 

• rsZ, tnot snbtTahendo 



17. BZ] ZB e corr. F. 18. viti] snpr» F. Sl. ZH] 
orr. m. 2 V. 22. Zvl] in m. P. erz] in rM. p. 
ia^Cv B. op*p3 om. P; oe&g ae<i V («pa eraa.). 21. 
r] r in ras, B. 26. tais Svai V. 



310 ETOrXEIflN e'. H 

reCvovSav vito (liav %wv (Gav yavimv " xk\ tieg kot- 
jcag BLQOL nXsvQag zatg i.otnatg alBVQatg ieag b^bi ' ItSti 
aqa tj Z& xu&etos t^ 2.K xa&^Tm. o^otras Sij ^ffj;- 
■&i}<j£rai, ort xai fxcerij t<5v ZA, ZM, ZH ixaviQa 
5 rav Z&, ZK tCt] iariv aC aivte apa Bv&itai m 
ZH, Z@, ZK, ZA, ZM itsai allilais tlaCv. 6 «pa 
xivTQSt T<p Z SiaOT^^iiaTi di ivl Tmv H, &, K, A, M 
KvxXoq yQatpofiBvog jj^bi xal dia tav lomtiov ernitimv 
xal irpdil^BTai Tmv AB, BF, F^, ^E, EA Ev&simv 

10 dta To off&^icg Btvai raj JiQog totg H, &, K, A, M 
ffrj^Btoig ytDviag. bI ya^ ovx iipd-^BTat avToav, ttXf-a 
TBiiBt avTag, 0vn^^6ETat ttjv ifj Sia(iitQa> ToiJ xvxXov 
repog og&ag an axffag dyo^ivijv ivTog xiittBiv tou 
xvxXov S^rep aTOJtov idtix^' ovx apa o xivTQ^ r^ 

15 Z StaGT^iittTi Sb ivl tav H, ©, K, A, M aijfieitov 
ygatpofisvog xvxXog Ttfist Tag AB, BV, F^, AE, 
EA Bv&sias' itpdiliBTttt apa avtiav. yByQd<p9oi tog 6 
H&KAM. 

Eig ttffa To So&iv JCSVTuyavov , o iffriv itSOJtkev- 

20 Qov T£ xal laoycivt.ov, xvxXog iyyiyQaicTai' oirep idst 
jtoiijffat. 



HbqI t6 So&iv nBvrdyavov, o ifftiv iao" 

nXBVQOv TE xat ieoyoivtov, xvxXov XEQtyQaii^ai. 

"Efftai ro So&iv JiBvtayavov, o istiv IffoxXevQov 

■> T£ xal ieoydvtov, t6 ABTJE- Set Sii ntffl to ABrJS 

ittvtdytovov xvxXov xsQtygdtpai. 

4. ZH] MH P. 5. hxiv ta>i V. 7, H] m 

Ze. ZK. ZA, ZM Gregorius. 10. M} om. P. 11. ai}^tt- 

ois] om. Bp, 12. «!]»] ij Bp. 13. dyop.ivti Bp, 14. 

kStiii^ij] otu. Bp. 16. Ktti Sitfffr^finri fV Bp. Zff, 28, 



9* 

QOV 



ELEBIENTORUM LIBER IV. €11 

iQB. itaque etiam reliqua latera reliqois lateribus 
ilia habebunt. itaque 29«« ZK. similiter de- 
itrabimaSi etiam singulas rectas ZA, ZM^ ZH 
06 ZBj ZK aequales esse. itaque quinque rectae 

ZBf ZK, ZA^ ZM inter se aequales sunt. ita- 
}ui centro Z radio autem qualibet rectarum ZH^ 
ZK^ ZA^ Z M describitur circulus, etiam per re- 

puncta ueniet et rectas AB^ BFy FA, AEy EA 
ngety quia anguli ad puncta H, 0, Ky Ay Mpo- 
recti sunt. nam si non continget^ sed eas secabit, 
[et^ ut recta ad diametrum circuli in termino per- 
icularis ducta intra circulum cadat; quod demon- 
um est absurdum esse [III^ 16]. itaque circulus 
*o Z radio autem qualibet rectarum ZHy Z9, 

ZAy ZM descriptus rectas AB^ BT, rA, AE, 

non secabit; ergo eas continget. describatur ut 

KAM. 

Srgo in datum quinquangulum, quod aequilaterum 

.equiangulum est^ circulus inscriptus est; quod 

tebat fieri. 

XIV. 

^cum datum quinquangulum, quod aequilaterum 
equiangulum est, cireulum circumscribere. 
Sit datum quinquangulum, quod aequilaterum et 
liangulum est, ABFAE. oportet igitur circum 
FAE quinquangulum circulum circumscribere. 

ZA, ZM sv&smv Gregorius. 16. xvxZos] m. 2 V. 
/cy^aqp^d^o) mg] xat iati iyysyQafifiivog dtg in ras. m. 2 F. 
9KAM] in ras. F; litt. H& q corr. m. 1 p. 20. yi- 

rxat V, imyiyQantai. F. 24. o iativ] om. Bp. 26. 

«ytavov] mg. m. 1 F. 



312 ETOIXEIflN 3'. 

TfTrfiiJffdDj Sii ixartQa tmv vno BFJ, F^E yo- 
vtcov dC%a vno sxar^Qas rtov rz, ^Z, xal aim xov 
Z 6-^fieiov, xa&' o Gvii^dlkoveiv at tv&fiai, £id xi 
B, A, E tiiWiEla iitB%£v%9a6av svd^itai al ZB, ZA 
5 ZE. ofioiag 6r] ra %qo tourow tf£tj;d^ff£ta(, OTt ) 
ixdifTf] Tmv vito FBA, BAE, AE^ yavitov di^ 
tttfirjrai vno ixderiis tSv ZS, ZA, ZE ev&Eim 
«al ijccl teri iaxlv rj vao BTJ yiovCu r^ vito P^i^l 
xaC ieri, r^ff }t\v vno BF^ TjfiCdEia ^ vao ZT^, i^ I 

10 S^ vito rJE iifiCaiitt 57 vno r^Z, xal ^ ino ZTJ 
a(fa tri vno ZAV ioriv Istj- (offrc xal nKev^fa ii 
Zr nXevQa rfj Zz? iativ la-ij. oftoCmg 81] deix9^ 
eitat, OTi xal ixderrj rmv ZB, ZA, ZE ixariQa trfl 
Zr, Z/i ieriv terj' aC nivtc aga ev9etat at ZA 

15 ZB, Zr, Zd, ZE teat dX^Jiais eieCv. 6 apa xh 
t^Gi tm Z xttl Sittef^fittri ivl rmv ZA, ZB, ZF, ZA 
ZE xt'xAus yQaipofuvos ^isi xal dia rcov f.oiJiav O^ 
(uCmv x«i eerai nsQty ty QanfUvos- n£Qi'yByQdip9ca xa 
tVtra 6 ABTAE. 

20 IleQl aga to tfoS^lv nevrdyavov, o ietiv leoitke^. 
(fov re xal Ceoycoviov, xvxKos itepiyeyQomai' onaf 
iSei noi^eai. 

ECg lov So&ivra xvxkov iidyaivov laonABX 
25 pov re xttl Caoydvtov iyyQu4'at. 

"Eeta So&elg xvx^los o ABF^EZ- Set S^ » 
TOi' ABVJEZ xvxkov ildyavov iaoxXevQov re xi 
leoydviov iyyQa^at. 

l. BTJ] ABJ in ras. F, seq. aestig. iJ. 2. JZ] in rag. 
m. 3 7| JZ Pv&tiav F {ev9i(itv m.2 in mg. traneit). «"o] 
aoir. in twd m. teo. F. i. B, A, E] "A, '£, £'" F, 6. i^] 




ELEBfENTORUM LIBER IV. 313 

eeiur igitur uterque angulus BFdy F^E in biuas 

B aeqoales utraque recta TZ; JZj et a puncto 

quo reetae concummty ad puncta By Ay E du- 

r rectae ZjB, ZA^ ZE. iam eodem modo; quo in 

edenti propoedtione demonstrabimus [p. 308, 16]; 

dngulos angulos FBA^ BAEj AEA singuliB 

ZBj ZAy ZE in binas partes aequales diuidi. 

loniam L BTA — FAE, et L ZTA — % Br^, 

'Z — y, TAE^ erit etiam L ZFA — ZAT. quare 

L zr^^Z^ [I; 6J. similiter demonstrabimuS; 

etiam singulas rectas ZB^ ZA^ ZE utri- 

que rectae ZFyZA aequales esse. itaque 

quinque rectae ZA^ ZB, ZF, ZA, ZE 

inter se aequales sunt. quare qui centro 

^ Z et radio qualibet rectarum ZA, ZBy 

ZA, ZE describitur circulus, etiam per reUqua 

;ta ueniet, et erit circumscriptus. circumscribatur 

it ABTAE. 

Srgo circum datum quinquangulum, quod aequi- 
*um et aequiangulum est, circulus circumscriptus 
quod oportebat fieri. 

XV. 

[n datum circulum sexangulum aequilaterum et 
liangulum iuscribere. 

Sit datus circulus ABFAEZ. oportet igitur in cir- 
im ABTAEZ sexangulum aequilaterum et aequi- 
ulum inscribere. 

\. %ai] om. Bp. 7. ZB, ZA, ZE] Pp; ZA, ZB, ZF 
'era8.)F; BZ,ZA,ZE BV. 9. iattv P. 16. Zz^,ZE]oin. 
jorr. m. rec. 16. xa/] comp. insert. m. 1 F. dl M P. 
&ifa]PY et F, sed punctis notat.; om. Bp. Sod^hv uQa 
in F ai^a insert. m. 2. 24. xvxAo F. 27. iidyoivqv] mg. F. 



314 ETOIXEIfiK 3'. 

xal ilXri<f&a ta xsvtQOV tou xvxlov to H, xal xiv- 
tpp ^Csv rp A SiaStijfittti Si za AH xvxlog yBygdg)- 
»a 6 EHr&, xal ini^evx&Biaai at EH, VH dtjJz" 
6 ^siaav ial ta B, Z e^ftttu, xul ins^Evx^aattv at 
AB, BF, r^, ^E, EZ, ZA- Xiya, ozt zb ABFAEZ 
B^dyavov titoxkevQOv rs iati. xaX leoyaviov. 

'Easl yiiQ ro H ffjjftEtov xivtQov ietl tov ABVAEZ 
xvxXov, iaij ietlv ij HE v^ HA. ndkiv, insl xo A 

jo ujjficrov xivtpov ietl xov Hr& xvxXov, Hotj iazlv 
T] AE ty AH. akX' ij HE zij HA idsix^j} [gi}' Mcl 
5) HE apa z^ Ed l'6ri i<SziV lConXBVQOv aqa iaxl 
xo EHA tQiyavov xal al tQttg apa avtov yiovi«i 
at vTto EHA, HAE, AEHteat dKl^kaig sleCv, iasi- 

15 di^xBQ zav iaoOxBlmv ttftycovmv at JiQog xy ^daBi yet- 
vtai teat aXl.i]Xatg bIgCv ' xai sletv at rpEtj xov tfft- 
yavov yavCai Svalv oQ&atg teaf tj aQa vito EH^ 
ytovia rp/ror ^iJTt Svo oQ&tav. hfioCiog Si] SBix^aetat 
xal ^ vito AHF TQtzov Svo oQ^mv. xaX ixel rj PH 

SO ev&Bta ial xijv EB eza&stga tag icpB^^g yavias zicg 

lOTo EHV, VHB Svelv og&atg teag iroifr, xal Xoix^ 

apa i] V7tb FHB XQttov iazl Svo op&iov aC aga 

vito EHA, AHF, FHB ymvCai teat aXX^^Xai.g eCaiv 

^ Saxs Xttl at xata xoQvipijv avtatg at wto BHA, 



1. ABTJ B. J^] e corr. m. rec. F. 2. H] poei raa. 
1 litt. r. 8. zf] non liquet ob ras. in F. nJH] d e oorr. m, 
rec. P- 4. EHr&\ o corr. m. rec, F. ijti£tux#»fl<H P 

corr. ta. 1. B. &] in ras, m. 2 FV. 6. Poat Xi-^ta add. H^ 
m, rec. F. 8. AEF^ Bp. 9. J] E P. 10. HVB] Pj 

HSK F; EHre BVp; in V seq. rfts. 1 Utt. 11. JEl E-J 
P, dU] SH F. dXU P. 12, Sga] m. 2 V. htiv 

ftfi) Vp. ^<Trit ieziv PF. 16. lconXtvQav F, aed corr, 
of] Bi rpits «r F. 18. flnlv} elaC V, nuI t^eiv] om. : 



i 




£LEM]SMTOBUM LIBER IV. 315 

ncatiir cirouli ABF^EZ diametrus A^^ et su- 
mfttor H centrum circuli, et centro A 
radio autem AH circulus describatur 
EHrSj et ductae EH, FH ad puncta 
B^ Z educantur^ et ducantur AB^ BF^ 
rd^ dEy EZj ZA. dico, sezangulum 
ABFAEZ aequilaterum et aequiangu- 

_ lum esse. 

Gun quoniam punctum H centrum est circuli 
""AEZy erit HE — HA, rursus quoniam A punc- 
centrum est circuli HFB, erit AE = AH sed 
nstratum est, esse HE^ HA, itaque etiam 
>■ EA. itaque triangulus EHA aequilaterum est. 
) etiam tres anguli eius EHA, HAE, AEH 

se aequales sunt, quia in triangulis aequicruriis 
li ad basim positi inter se aequales sunt [I, 5]. 
^es simut anguli trianguli duobus rectis aequales 

[I, 32]. itaque L EHA tertia pars est duorum 
rum. similiter demonstrabimus, etiam L AHF 
im partem duorum rectorum esse. et quoniam 

FH in EB constituta angulos deinceps positos 
!^, FHB duobus rectis aequales efficit [I, 13], 
i reliquus L THB tertia pars est duorum recto- 

quare anguli EHJ^ AHF, FHB inter se ae- 
)S sunt; quare etiam qui ad uertices eorum sunt. 



m. rec, sed daiv eras); aXXd p. 17. tcai sia^v Bp. 
of^a rij sed 17 del. m. 1 F. 18. tQitov] tarj 9. 19. 

'] r in ras. p. tgitov P. 20. arad^siaav, sed v del. 
22. tQ^tov P. iativ PF. 24. at] om. B. avtag 

rvrais B. 



316 ETOIXEiaN 4'. 



] 

stat I 



^HZ, ZHEiaai. tielv [zatg vno EH^, ^iHT, FHj 
at i'| &Qa yavitti al v%o EH^, ^HF, FHB, BB- 
AHZ, ZHE ieat aXl^^Xaig ttaiv. at S\ 'Ceat yatvi 
inl ifftav itsQiptQBimv ^tfi^xaeiv ' aC ^| «ga atQup^^fSi 
5 aC AR, Br, rj, ABt EZ^ ZA laai aAAijAats «'«''v. 
vnh b% xag ieag xsQKpEptias <^t tfSai iv&tiai mtotsi- 
voveiv at ?| uqa ev&stat i6ai dXXrilats eieiv la6- 
TtXcvQov apa ietl t6 ABVAEZ i^dyvivov. Xiyea ffij, 
ott aal iaoymviov. iittl yuQ i'6jj ierlv i] ZA ntpi- 

10 (pSQlia rfj EA jtBQtfpiQtia, xoivij HQoaxtie&a ij ABFA 
iciQiq>iQtia- oXrj apa i/ ZABFA oXy ty EAFBA 
ietiv iari' xul fii^ijxtv inl n%v t^g ZABFA tcsqi- 
tptQtiag i} vito ZEA yavia, inX S^ i^g EATBA 
jttQitpBQSias ij vito AZE ytovia' tatj affu rj vxo AZE 

!5 yaviu tfj vito /lEZ. oftoiiag dri SBix^Snai, ot( xofi 
at Xomal yaviat roii ABTAEZ i^uydvov xuta (liav 
ieat tielv ixatipa rav vmo AZE, ZEA yavtiav Coo- 
ycivtov UQa ietl ro ABFAEZ i^aycovov. iSeix^ 
Sl xal ieonXtvQov xal iyyiyQaxtat ttg tbv ABFAEZ 

20 xvxXov. 

Eig apa tov So&ivta xvxXov i^dytavov ie6xXfv$6m 
if xal ieoyiaviov iyyiyQaxtat ■ oatQ iSti Jtoi^aai. V 



1. Haai al.X>^luig V, seil dllrjlais del. m. 2; habet ed. Ba- 
ail. tiaiv] elai BVp. rais ujto EHJ, aHF, rHB] mg. 
m. 2 V; om, ed. Ba.eil., Anguatus. EH.d\ J e corr. F. 
Poat ^HF ras. S litt. V. 2. at f£ — 3. i).XT,late ilaif] mg. 
m. 2 V, om, ed. Basil- 4. at ?! aga] in raa. ra. 2 V. 6. 
EZ] EZZEZ P, sed corr, m. 1. 6, Si] aupta m. 1 F. 

at] om.V. Post to»(iiit F mg, m. 1; at AB, BT, Fd, aE, 
EZ, ZA; idem coni. Augustue. S. lari] om. Bp. Aij] 

supra m. I P. 9. yae] poatea ioaert. in F. ZA] PF; AZ 
BVp. 11. ZABr.J] pro B in P m. 1 est Z; corr. m. 2. 

Seq. in F ncfififfia anpra acr. m. I. Post EJTBA in F 



irii 



ELBMiarroBUM liber IV. 317 

jtHZj ZHE aequaleB siint [I, 15]. itaque 

;iiU EH^, JHr, rHB, BHA, AHZ, ZHE 

) Mqnalea sunt. aequales autem anguli in ae- 

B aiealras consistunt {JH, 26]. itaque sex 

iB, Br^ rj, jJE, EZ, ZA inter se aequales 

A rab aequalibus arcubus aequales rectae sub- 

t [lll, 89]. quare sez rectae inter se aequales 

ergo sexangulum ABF^EZ aequilaterum esi 

idem aequiangulum esse. nam quoniam arc. 

BJ^ communis adiiciatur arcus ABF^, itaque 

rj «- EJFBA. et in arcu ZABFjd consistit 

dj in E^FBA autem arcu LAZE. itaque 

LAZE^jdEZ [HI, 27]. 
«r demonstrabimus, etiam reliquos angulos sex- 
L ABFdEZ singulos aequales esse utrique an- 
dZE^ ZEJ. itaque sexangulum ABFAEZ ae- 
guhim est. demonstratum autem, idem aequilate- 
ssse; et in circulum ABFAEZ inscriptum esi 
1^0 in datum circulum sexangulum aequilaterum 
quiangulum inscriptum est; quod oportebat fieri. 



8Cr. m. 1: nsQitpBQtia, 12, ZABFJ] seq. ras. 1 litt., 

ras. V; £ postea add. Bp. 14. AZE] JZE F; corr. 

16. JEZ] ZEd P. Post nai in P del. s m. 1. 

EJ] i/EZ F. 18. i0Uv F. 



STOIXEmN 3'. 



ytovov n^evfff^M 



'Ex d^ TOiJrou (pavtQOV, oti ij tou iS^aymvov ji 
rejj ^ffrt rg ^x Tov xivTQOv tov xvxXov. 

'OfioCtaq 6i totg inl toi ^iivraycovov iav Sia ttov xaza 
6 Toi' xvxlov SiaiQeeamv itpanzo^ivas tov xvxlov dya- 
ycanfv, XEgiyQuqiiqaaTai. xegl rov xvxkov t\ayfavov 
LOonXsvQQV T£ xal ieoymviov dxokov&ios tolg ial rov 
jiEvtaytovov eiQiniEvois. xal i'tt 6t.tt tcov ofioiav xoCe 
inl Tou ixEvzaycovov tigi}{iivois slg tb So&iv i^dytovov 
10 xvxkov iyyffttiltonev te xal atpi,ypd^ontv oxtQ ISei 
notijeat. 



Elg zov So&ivxa xvxXov jievttxaiSexdyoivai 
itf6xi.tvffov i£ xal iaoyatviov iyygdtltai. 
■) "Eeta 6 So&tig «iIxAos o ABr.d- 3ii Si] ilg i 
ABr/i xvxXov xtvztxatStxdyavov laonXtvQov tt i 
iaoyaviov iyygtttliai. 

'EyysyQKip&co lis tav ABT^} xvxlov ZQiyavov fkkv 
iaonXfLQOv rou tiq ctvtov iyygaq)0[iivov xXtvga ^ 



XVnopiOfw. Simplicius in phjB, fol. 1&; cfr. p. 319 not.! 



1. jcdgiofio;] m. a V. 3, iatC] om. p. 4, ofioiaie — 
jtypnt/iouev] non habnit Campanue; sed n.p.330,148 
. D(Loicas St zoh ^Jil Toi Tcivzaymvov'] P; xa/ Theon (BFVp). 



Iv 

t 



iriotyenWouev] non habnit Campanue; sed n. p. 330,14sq. 
■ ' ' •• - ItiI iou :i*viovri»oi.] P; xa/ f heon (BFVpl 
K littHpEceon'] P; /i, B, P, d, E, Z ariiirlaiv 



Theon (BFVp); r in raa. V. 5. i6v] Boripai; 

^qpunTO^ev.e B. Ante dyayta^fv in F add. S (in fin, lin.) v 
(iii init. eequentifi), 8. ofioiaiq Bp, 10, xvnlov] siipra m. 

1 F. lE val lEEpiyea^ofifv] om, P. omi/ ISu nDi^oai] 

mg. F, in qao omisao numero quattuor prima uerba pro^. 16 
cum antecedentibus coniuncta snnt, ita ut Fl pro litt. imtioli 
rit; poetea corr. m. 1 uol 2. 13. TcevTaKaiStKdyaivov P, nt 

lin. 16. 18. lyytyi/dtpituJPF ; ytygdifiS-ia BVp; lvj]fii,6a&<o 

AngUHtnB. 19. loii] om. P. Buid»] corr.ex ouio' m. 1 F. 



ELEBIENTORUM LIBER IV. 319 

Corollarium.^) 

c manifeatam est^ latus sexanguli aequale esse 
irculL 

eodem modo, quo*) in quinquangulo, si per 
diuisionis in circulo posita rectas circulum 
entes duzerimus, circum circulum sexangulum 
lierum et aequiangulum circumscribetur secim- 
y quae in qoinquangulo explicauimus [prop. XII]. 
iterea simili ratione ei^ quam in quinquangulo 
oimus [prop. Xm—XIY], in datum sexangulum 
m inscribemus et circumscribemus; quod opor- 
ieri. 

XVI. 

datum circulum figuram quindecim angulorum 
.teram et aequiangulam inscribere,^ 

datus circulus ABF^. oportet igitur in ABF/i 
m figuram quindecim angulorum aequilateram 
oiangulam inscribere. 

cribatur*) in ABFjd circulum ^Flatus trianguli 
iteri in eum inscripti [prop. II] , et ^^ latus 

Huc refero ProcH uerba p. 304, 2 : x6 61 iv xm SsvxiQfp 
%8£(i6vov (sc. noQiafia) nQopXrniccxos ; nam oum neque 
, 4 noQ,, quod theorematis est et insuper subditiuum, 
lent neque cum alio ullo — xo enim ostendit, in eo 
de quo agitur, unum solum corollarium fuisse — , pro 
soribendum d'y b. e. xsxccQxa), binc sequitur, Proolnm 
noQ.] pro corollario non babuisse. 

Mutauit Tbeon, quia cum lin. 7 8q. synonyma esse pu- 
quod secus est; dicit enim: si ut in quinquangulo con- 
les duxerimus, eodem modo demonstrabimus cet. 

Cfr. Proclus p. 269, 11. 

*EyysYQd(p9^(o ideo ferri posse uidetur, quod latus tri- 
in circulum aptamus triangnlum inscribendo. 



320 sToixEiaN e'. 

AFy XBVtaymvov dl laonkBvqov {j AB' oXiav «0 

iotlv 6 ABFiJ xvxXog Hcav Tinjiiaroiv Stxajiivi 

toiovTiai' T] [ilv ABP 7tEgnpiQ£ia tqCtov ovsa ■ 

xvxXov letai Jtivzs, ij di AB nsQi^iptia nifirov c 
5 rotJ xvxlov iarai rpiiDV Xoini} «pa tj BF t^v fa^ 

Svo. reT^rjS&as tj BF SCxa xata to E' ixaTi^a t 

Tmv BE, ET «tQi^BQsmv JttvTExatdixarov ieti rvv ' 

ABFi^ xvxlov. 

'Eav ttQa iTtilEv^Rvtes tag BE, EF Cdag avtatg xatct 
10 To Svvtxis i^ii&siag ivaQiioOaftsv stg roi' ABr^[E] 

xvxlov, israi eig avrov eyyeyQaii(iivov TtsvttxaiSexa- 

yavov lOoTtXsvQov rs xal iOoycoviov " ossg iSsi sroi- 

ijeai. ^ 

'OpLolmg S\ rotg iitX 
15 Toti nevTctymvov iuv Sia 

Tiav xara tov xvxlov 

StaiQiesaiv itpantofii- ■j.f 

vag TOU XvxXov aya- 

yai[isi', TCSpiyQaip^esTai 
negl rov xvxXov xevrs- 

xaiSexayiovov ieoaKev- 

(fov Ts xal ieoyaviov. 

iri Si Sitt Tmv oiioicav 

Totg ixl rov Tcevtayto- 
25 vov dsi^scov xai sig ro So^^iv 

iyyqa^onsv Ts xal aeifiY(/aii>Q[iev 




5. ftnai] -at in raa. V. agti] om. P; m. 2 Y, Bnpn 1 
BT] r in ras, F. 6. 8vo] f P. 7. hti] om. Bp; tm 
P. 9. ET] P; ET tv9n'as Theon (BPTp). avxuts] ooir. " 
ei auiae m. 2 B. 10. ABF^ p, ed- Baeil. 11. nivTcxai- 
dtiiayBivov] mg. B. 12. noi^aai] iii^at BTp. 14~~S6 

habuitC&mpanuslT.ie. 16. tov] ota. P. 18. tov] laf lov F. 



ELEBfENTORUM LIBER IV. 321 

ingoli aeqmlateri. itaque si ABFJ circulus 

Qim partibus aequalibus aequalis ponitur, earum 

e aequalis erit arcus ABr^ qui tertia pars est 

» arcus autem AB^ qui quinta pars est circuli; 

itaque reliquus arcus BF duabus partium ae- 

n aequalis est. secetur arc. BF m duas partes 

S8 in £ [niy 30]. itaque uterque arcus BE^ 

linta decima pars est circuli ABFJ. itaque si 

reotis BE^ EF semper deinceps rectas aequales 

mlum ABFd aptauerimus [prop. I], in eum 

»ta erit^) figura quindecim angulorum aequilatera 

uiangula; quod oportebat fieri 

dem autem modo, quo in quinquangulo, si per 

. diuisionis in circulo posita rectas circulum 

gentes duxerimus, figura quindecim angulorum 

itera et aequiangula circum circulum circum- 

;ur [prop. XII]. et praeterea per demonstrationes 

3 iis, quibus in quinquangulo usi sumus^ etiam 

tam figuram quindecim angulorum circulum in- 

mus et circumscribemus [prop. XIII — XIV]; quod 

$bat fieri. 



Aeqnilaterom fore figuram inscriptam, patet. tum ean- 
equian^am esse, simili ratione demonstrabimus , qua 
\i Enclides p. 316, 9 sq. — memorabilis est in bac pro- 
ae usus uocabuli xvxXoff, quod contra I def. 15 pro nBqi- 

ponitur (p. 320, 2. 4. 6. 8.). 

] in ras. V. $f\ m. 2 V. rmv oikolmv] corr. ex t6 

m. 2 B. 26. xai] postea insert. F. Post nsvtS' 

dymvov add. Tbeon: o iativ loonXsvQOv ts xal laoycoviov 
p; iatt p), sed cfr. p. 318,9. 26. iyyQd^ipa^iisv P. 

uipmiisv P. onsQ iSsi notrjaai] P; om. Tbeon (BFVp). 
ine: EvtiXstdov atoi,%sCaiv 8' PeiB; Ev%XslSov atoi%sCmv 
sa)vog indoasoig 8' F, In fig. tf P, is' F. 



cl def , edd. Hciberg et Menge. 21 



APPENDIX. 



Sl' 



DEMONSTRATIONES ALTERAE. 



Ad lib. II prop. 4. 

Aiya, oit ro ajto t^s AB xitQayavov teov iazl 
totg rs «rco tav AF, VB vszgaycivoig xal rm Slg 
vxo tdv AF, FB jtfpiejjo.uEVp OQd^oyavC^. 
5 'Ewl ya(f tijg avt^s xatayQatp^g, iirsl tat] iatlv tj 
BA tr. A^, tejj iSTl xal ytavia ^ v%6 ABjd tr^ 
v:i6 A^B' xal iTtsl xavtog tgiyavov ai rpefg yaviat 
Svalv oQ&atg taai sleCv, tov AAB «pa t^iyaivov al 
XQttg ymvittt at vno A^B, BAA, ABA SvsXv 6q- 

10 &alg tCai sCoiv. oQ&i] 6h ij vno BAd' kontal aga 
at vno AB^, A^B fua op&^ tOat- bIoC' xaC Eiai.v 
teat ■ Bxatiga apa rcav vitb ABd, A^B 7j[iC0Bitt iottv 
OQ&^g. opQij di ij vno BVH' tatj yuQ ieti tjj ait- 
EvavtCov Tij ng6g tp A' Xotiti] aga ^ vao THB ifitC- 

16 esid icttv 6p*^g' taij &^a ij vno FBH yavCa t^ v«6 
FHS- mets xal }ti.svQa i] BF tfj FH ieriv tOt]. dXX' 



Addidit Theon (BFVp); mg. m. rec. P; de Cunpano nfl 
p. m nok 1. |l 

1. xol teUiDS P. 3. m] ra. 2 p. AF} oorr. ex AB P.l 
6. BJ\ AS p. {atC] om. V. 7. i^eC] non Uquet in F. 
8. ttalPB xai AJB — 10. ftnCv] mg. m. 2 Vp. i.A^B] 
ABJ Pp. 9. AJS\ ABJ Pp. BAJ] AJB P, JBA p. 



n, 4. 

Aliter.i) 

co, esse AS^ — AF^ + FJJ» + 2 ^F X FB. 
m in eadem figura [p. 127], quoniam BA^AJy 
dam L^BjJ = A^B ^^5]. et quoniam cuiusuis 
;uli tres anguli duobus rectis aequales sunt, erunt 
knguli trianguli AJBy scilicet 

AJB + BAJ + JBA 
is rectis aequales [I, 32]. uerum LBAJ rectus 
itaque reliqui AB^ -^- AJB uni recto aequales 
et inter se aequales sunt. itaque uterque AB^y 
i dimidius est recti. rectus autem L BFH. nam 
alis est opposito, ei qui ad A positus est [tum 
31]. itaque reliquus L FHB dimidius est recti 
2]. itaque L FHB = FBH. quare etiam 

Br=rH [I, 6]. 

L) Haec demonstratio parum differt a genuina; nam prae- 
nitinm demonstrationis , qua ostenditur, FK quadratum 
cetera eadem. 



Q B-4z/ Pp. 11. slai] non liquet in F. xa/ slciv ftyatl 

F. 12. AJB, JBJ p. 13. ccTCsvcivtiag p. 14. tc5] 

ex To V. 15. FBH] THB P, F e corr., V sed corr., 

ymvCd] om. p. 16. THB] B, F eras., V corr. ex TBH 

; FBif Pp. aXXa p. 



326 DEMONSTEATIONES ALTERAE. ■ 

15 (i^v rS TTj HK iGriv lan, 7} 6i rHrjj BK' lao- 
nXBVQOv KQa ierl ro FK. ^jjei Sl xal op&^v t^v tijtd 
FBK yavCav zitQayavov upa iorl rh FK' xai ioxiv 
uao T^s FB. $i& TK avra Sii xal to Z& rtrgdymvov 
s ian, xai ieriv taov rm unb r^g AT' ra aga FK, 
®Z TirpayravK ieri, xaC iartv iaa rotg axh rmv Ar, 
VB. xal ixd taov istl ro AH ra HE, xal iari ro 
AH To vah rroi/ AF., FB' fffj; )'«p 1] FH rij FB- 
xal ro EH aga l'aov iarl ra vith rav AF, FB. ra 

10 «pa AH, HE taa iarl r^ Slg vab rwv AT, VB. Sart 
Sl xal ra FK, ®Z taa rotg «Jio rrav AT, FB. ra 
a^a FK, &Z, AH, HE iOa iorl rotg re aah rmv 
AF, FB xal rp Slg Wh t6v AF, FB. akXa ra FK, 
®Z xal rct AH, HE oXov iarl ro AE, o iariv aith 

IB r^S AB rstQByavov ro Kpa aao t^g AB tfrQayavov 
iaov iatl rotg re dith riBv AF, FB tstpaydvoig xal 
rm Slg vnh rav AF, PB ittpisxonivp oQ&oyavitp' 
oasQ iSsi dst^at. 



Ad lib. in prop. 7. 
"H xal owrog. ins^svx&ta ii EK, xa! i-xel h 
20 iarlv ii HE rij EK, xotv^ Si 73 ZE, xal ^aOtg ij Z 
§dasi rTj ZK Hati, ymvia apa ^ iisto HEZ yavia rjj 
vnh KEZ tat} larCv. aUa i) vno HEZ trt vnb ®EZ 
iariv tarj- xal ij vxh &EZ apa r^ '^^^ KEZ iativ 
fffi;, 15 ^Aarrmv r^ (isi^ovt " ojTf p iarlv dS^varoi 



I 



in, 7. Inflertam inter dSvvazov et ovx p. 182, 9 PBF7J 



1. Jimv] comp. supraacr. F. 2. vai] abBnmptni 
pergam.F. S. hup] lott td F. i. FE] BF Fp. 
6Z Pp. iaii TtTQdYtovov p. 6. /aii] hti* " "" 



DEMONSTRATIONES ALTERAE. 327 

rB = HK p, 34] etrH^BK [id.]. itaque 
ftjierum est FK. habet autem etiam L FBK 
1. itaque quadratum est FK] et in FB construc- 
flt. eadem de causa etiam Z9 quadratum est; 
[uale est AI^. ergo FK, 0Z quadrata sunt et 
ba sunt Ajy et rB\ et quoniam ^H — HE 
] et AH^AFX FB (nam FH^rB), erit 
EH = AFX FB. itaque 

AH+ HE = 2ArxrB. 
i etiam FK + eZ^ AF^ + rB\ ergo 

'ez+AH+HE=Ar^ + rB^ + 2ArxrB. 

'K + «Z +^H + HE = AE = AB^. ergo 

^B» = An + rjJ^ + 2 ATxrB] 
erat demonstrandum. 

in, 7. 

el etiam ita: ducatur EK. et quoniam 

HE = EK, 
E communis ost, et ZH = ZK, erit etiam 

Z. HEZ = KEZ [I, 8]. 
n L HEZ = 0EZ. quare etiam 

L ®EZ = KEZ; 
r maiori; quod fieri non potest [u. fig. p, 181]. 

7, Tco] x6 B et V (corr. m. 2). 6. hxi\ iariv F. 
] mg. m. 2 F. HE] EH B et FV m. 2. 8. vno] 

ex ano p. l'ar} iazl yccQ P. 9. EH] HE p. apaj 
P. vwo] dno P. 12. rjf ] om. F (ras.). HE] EH 

Ts] supra m. 1 p. 13. AT] FA F (prius). 14. AE] 
B. p. 19. mg. &Xl(og p. 20. HE] in ras. g), Eif p. 
EZ P. ZH] PF; JFfZ BVp. 21. ycovifa] om. B. 
ativ tari Bp. ctU.' FV. HEZ] corr. ex EEZ m. 1 

)rr. ex EZ P. 0EZ] ZE0 P. Post hoc uerbum in 
OQ. 2 insert. ycorta comp. 23. 0EZ] ZE^ P. 24. 17 

mv TJ yLhitavi] in ras. V. IXdaamv F. laxlv] om. p. 



DEM0N8TBATI0NES ALTERAE. 



1 



Ad lib. ni prop. 8. 
"if xccl a!,lais. Eaei,£vx&a Tj MN. insl fffij iarlv 
KM v^ MN, xoivii S^h ij MJ, xkI ^aatg ^ JK ^aae~ 
rij jdN HOTj, yaviK apa ij ujio KMid yavia. tj; hno 
^MN iaxw Ceij. aXX' 17 vao KM/d r^ vao RMJ 
6 iaxiv lei\' xkI Tj V7t6 BM^ aga xf) vxo NMjJ iaxiv 
tati, 1) iXaztav TJj (ifi^ovi ' oitig ietlv Kdvvarov. 



Ad lib. III prop. 9. 

'AXXas. 

KvkXov yap tov ABF {{Xt^ip&^m ti arjiietov i 

tb /3, ano S% tou z! ngog tov ABV xvxXov jtjmH 

10 ntTtziraaav nketovq ^ Svo taat ev&eiai 'aC 

^B, ^F' Xsya, OTi t6 Xtjpd-iv arjfietov t6 jd xivTffi 

ietl Tou ABT xvxXov. 

Mij yaQ, aXX' el Svvatov, i'Gtm ro E, xa) imtE 
&e[6tt 7} ^E iJnJj;*o iitl ta Z, H arjiieta. 
16 uffa StdnerQos ieri tov ABF xvxXov. izel ovv 1 
kXov tov ABF ial tijs ZH Si.a[iitQov eHXrptrai 1 
ayi^elov, o fti) io^Tt xivtQov roiJ xvxXov, t6 ^, ftsyiay 
[ihv Ssrat ij JH, iieitfav Sl ■ij p,lv AF rijs -dB, ■ 
Se AB T^s A A. aXXa xaX fcij" one^ iatlv aSvvatm 
20 ovx Kpa ro E xivXQOv iarX Toi; ABFxvxXov. oiioim 



III, 8. loBertnm iater ISeiz^Ti et ov* p. 188, 20 in i 
111,9. Poat genuinam PBFVp; om. Campaims. 

l. ivcl ovv p. 8. M^] am B. 3. lattv foij p. 
KMd\ K^M F; corr. m. 2, iavla\ om. p. 4. 4MA 

NMJ P. rarj latlv BV; i<sxi fffij ip. atk& P. 



DSM0N8TRATI0NES ALTERAE. 329 

ra, 8. 

etiam aliter: dacator MN. quoniam 

KM = Af iV, 
oommmiis est, et ^K = ^N, erit 
L KMJ — JMN [I, 8]. 
L KMzl = BM/1. quare etiam 

LBMJ^NMJ, 
maiori; quod fieri non potest [u. fig. p. 185]. 

m, 9. 

n intra circulum ABF sumatur punctum ^, 
/ ad circulum ABF plures quam duae rectae 
3s adddant A^y ^B^ ^F. dico, sumptum punc- 
^ centnim esse circuli ABF. 
sit enim, sed, si fieri potest, sit E, et ducta 

^E producatur ad puncta Z, H, 
ergo ZH diametrus est circuli 
ABF. iam quoniam in circulo 
^^F in diametro ZH sumptum 
est punctum quoddam ^, quod 
non est centrum circuli, maxima 
erit ^H, et 

^r>^B,JB>^A [prop. VII]. 
i etiam aequales sunt*, quod fieri non potest. ergo 
am E centrum circuli ABF non est. similiter 

Bupra 8cr. comp. m. 2 BF. 6. iXdaaoDv Fp. ictiv] 

7. aXXayg] mg. m. 1—2 F, qui in mg. habet i\ sea 

In B ante SXXoag ras. 1 litt. 8. Post yuQ ras. 5 litt. 

10. taat] supra m. 2 F. sv&eion taai V. AJ] PBF; 

corr. m. 2 V, pg?. 12. iat^] om. B. 14. Z, H] H, 

16-. iati] iativ FV. 16. Post ^BF in P del. kv- 

xijg] g eras. F. 17. crjfisiov t6 J P. t6 J] om. 

18. iatai] in ras. m. 2 V. 




330 DEMONSTRATIONEB ALTEHAE. 


1 


dij dai^Ofisv, ozi ov6' «AJo n nXijv Toij ^' 
«pa fff)[i£lov xivt^ov iaxl xov ABF vvxlov 
ISii. Sitin. 

5. 
Ad lib. III prop. 10. 


ro ^ 
inig 

■ 


1 


'Jinas. 





5 KvxXos ycp nakiv o ABF xvxkov tov ^EZ xsft- 
vtTca xara nlsiova 6t]n.Ela ^ i5t!o t« B, H, &, Z xal 
Eikrifp^Ki To xivTQOv rov ABrxvxlov xo K, xal int- 
tivx»fa0av at KB, KH, KZ. 

'Eitsl ovv xvxkov rou jdEZ efAjjirrwif %i QTjfittov 

10 ivzog To K, Xttl ano roii K ngog tov AEZ xvxXov 
jtQoSntmcSxaei nlslovg ^ Svo fffet EV&sHai aC KB^ 
KZ, KH, xbKaQa ffijftefoi/ xivtffov iGrl tou ^EZ 
xvxkov. tffrt di xal Toi) ABT xvxkov xsvt^ov to K' 
Sva aga xvxktav TB^vovtfov akk-^kovg xo avrb xivxQov 

15 iorl xb K' o«£Q iotlv aSvvaTov, ovx a^a xt<xkos xvxkov 
xi(ivsi. xata nkEiava Gtjfisia i} Svo ' onsQ iSai Sti^ai. 



6. 

Ad lib. III prop. 11. 

'Akka Sii jttnxita mg v HZr, [xal'] ix^e^^ai 



ni, 10. Poat genmnain PBFVp; om. CampanuB. 
111,11. Poat gennitjam PBFVp; non habet CampamiB. 

1. oi!*f V. 2, OJife ISei Sei^tti] Pp; :^ B; om. FVlj 
ifl' mg. F, Bed eras. 6. S, Z] Z, BVp. 9. JB; 
raB. V. Tt] m. 2 P. 10. fvzoi] om. P. 11. npd" 

tv»eiai faai V. 12, KZ, KH] KH, KZ 
I m.l; corr. m. 2. ai>a K F. 13. ?orw P. H. 
P; corr. m. tec. 16. iaziv] om. p. 16. litiiiH} 



^lW 



DEM0NSTRATI0NE8 ALTERAE. 331 

demonstrabimas^ ne aliud quidem ullum centrum esse 
praeter z^. ergo ^ punctum centrum est circuli ABr\ 
quod erat demonstrandum. 

m, 10. 

Nam rursus circulus ABF circulum AEZ inplu- 
ribus quam duobus secet punctis B,Hy0,Z, et sumatur 
centrum circuli ABF et sit Ky et ducantur KB, KH, 



iam quoniam intra circulum ^EZ sumptum est 
punctum Kf et a JT ad circulum ^EZ plures quam 
duae rectae aequales ad circulum JEZ adcidunt KB, 

KZ, KH, punctum K cen- 
trum erit circuli jJEZ [prop. 
IX]. uerum K etiam circuli 
ABF centrum est. ergo duo 
circidi inter se secantes idem 
centrum habent JT; quod fieri 
non potest [prop. V]. ergo 
circulus circulum non secat 
in pluribus punctis quam 
duobus; quod erat demonstrandum. 

m, 11. 

Uerum cadat ut HZF, et producatur FZif in di- 
rectlim ad ® punctum, et ducantur AH, AZ,^) 




1) Haec demonatratio casus alterius post genuinam parum 
necessaria est. 

tBpLBi F; om. p. tsiivsi arnisia p. ij dvo] supra m. 2 Y. 
17. &Ua)g add. Vp, mg. m. 2 F. Post d^ ras. 2 litt. F. 
ri] supra m. 2 V. HZF] litt. H in ras. F, om. p; r in 

ras. p. xa^] om. P (F?). nQoas^§s§Xria&a) BVp (F?). 



332 



DEMONSTRATIONES ALTEEAE. 



&aieav at AH, AZ. 

'EatBl ovv alAHjHZ [leitovg siel tijs AZ, dXka ij 
ZA ierjliatl^Ttj Zr, rouriffrt TijZ&, xoivij dg>tiQ'^e9(a 
5 T] ZH' Xom^ «pa ^ AH Aoisr^s tijg H@ (iEit,B3v ietiv, 
tovtiettv 7} HzJ trjq H®, Tj iXaTTmv rijg fiti^ovog' 
oxsg iOTlv aSvvatov. oyLoiag, xav ixtag ^ TOt' (ii- 
xpov To xivtQOv rou jisi^ovog xvxXov, Sei^ofiev [ro] 
cToiroi'. 

7. 
Aa lib. m prop. 31. 
10 "AlXas 

ij aTtodei^ig ^ow op&ijv elvai tijv vnb BAF. 

'Eml SmXij ietiv i] v:co AEF r^g v%o BAS 
fiJjj yap SvOl tatg ivtog xal anevavriov iari Se xigj 
ij v%o AEB Stnli] t^s ■vjiq EAT, aC apa vxo AEA 
15 AEF SiJtXaeiovig eCai r^g vno BAF. AXX' aC i 
AEB, AEF Svelv oQ&atg Heai sieiv 17 «pa tinro SA^ 
ogd^ ieztv' oxsQ i'dei, Set^ai. 



m,31. InBert. p.246, 2 post 9(t|i 






1. ^i. 
. PP. 



i^ 




raa. P. HZF P; FHZ B. 
akX' P. 4. ZA] PF; AZ B 

F. tnj T^s B. Zr] PF; rz BVp. 

5. fBTi PBV. 6, IXdaaiov Pp. 7. i<ni 
miaB. V. 8. rol om. P;_ corr. in . ' ' 
avta p. 9. moTtov] atotiiatefov 

Sc^ai P. 12. ,*£r] corr. ex E^ 

14. EJr]AEr F; corr. m. 2. 15. daiy P. B«a P. 
IT. DnEp fd'» dfi^ai} in mg. transit tp. 



DEMONSTRATIONES AU^ERAE. 



333 




iam quoniam jiH + HZ> AZ [I, 20], uerum 

Zj4 «-= Zr, h. e. ZA = Z©, sub- 
trahatur, quae communis est^ ZH, 
itaque AH> HS, h.e. HJ>He, 
minor maiore; quod fieri non pot- 
est. similiter, etiam si centrum 
maioris circuli extra minorem fu- 
erit positum, absurdum esse de- 
monstrabimus. 

m, 31. 

Alia demonstratio, angulum BAF rectum esse^) 
[u. fig, p. 243]. 

quoniam L AEF •= 2 BAE (nam 

AEr.= BAE + EBA\l,m^\ 
et etiam L AEB — 2 EAF [id.], emnt 

AEB + AEr=2BAr. 
nerum AEB + AEF duobus rectis aequales sunt p, 
13]. ergo L BAF rectus est; quod erat demonstran- 
dum. 



1) Cfr. Campanus 111,30. 




U llb. IV-TI 

* r iib. VII— X 

I II lib. XXI- XXI 11 
ilUb. XXIT— XXVI 




L JbrU. 's 'voll, ', 



Mt. dllMta* «d. MrrM —.«0 

litlBl XII Bd. nw«w< s.w 



PonpoBtl HdlM da chonignptaU 

llbrl tiH tM. ^rloi. . ... 

rorphjrldi sd, Wmr ,..,.. 

ParphjrrlMi »1. Sanek 

rtootai ad. FritdMn i 

1'rapartl» ed. MSUtr - 

Uvlitlllu «d. finrHU. I TnlL. , 

Ub, X (d. Balm . . . . - 

<lilnti> SnrrBMia nd. Xo^^lii. - 
Hariin BalBratlM wrl»l. «Ir*se1 

KbnlareaUiHcieiSiMi^l. s^.oll', ■ 
tlBkBlitRll eloglan HrBuUrhuill 

•d *v™ - 

Bitltlua NiinBUBBBi wt. im>r. . -- 
MBlluntlilii ad. J>»»Dit. Bd. IT , .- 
Kcrifl. hlBlor. ABEUBt*». > •ulL : 
Serlpl. nttrlci (iricrl gd l^xi^. 



8*BMa rlietsr i 

8t«o SRthBii ed, Lanalmii .... 

SapHarln (d, Dnikr/. Bd V. . . 

aiBiiolBs Btaoiii . h - 

8or«BBi ed. Hon 

KtBtlai od, (ju.ol. t vuU. .... 

- — Yul. 1 SilTBO od. Aulniii . . 

Tol n Fuio, L AoUUoU od. 



nartloKlBM id. Miinih: 

L Uiitett. i tdU.' . , . 



«>n»atB, AgrlflOtB o 

dlBlogBI «d. B«ti« 

TfrCBtlna ad. fMm'*'!! 

ToaUBentiiBi, boibm, flrWFC w! 



TheodonB FradroaB» ed. Htniur 

ThODphrHlBI id. Iflu.nirr, 3 TDll, 
ThFOphnuti (h«. xL Fafi. . . . 
Thacidldes od, ttMimi. 1 TDlL , 

Tlbnllni ed. UHIIm- 

UlplBB *], Ba-Me. Sd. n. . . . 
TalerlB* FtiecB* ed. Bta/irtv . . 
TiterlBB XulBiai ed. ffaliN . . , 




Schnl-WOrterbucher der klassischeii Sprach^ 

im Verlage vod 

B. 0. Teabner in Leipzig. 

Qrieohieoties SohulworteFbuoh. 2 Bande. gr. Lex.-a. 



IiateitusobeB SohulworterbTich. Vou F. A. Heinio] 

2 Bde, gr. Lei.-8. geh. 



Spezial-WorterbQeher. 
Worterbuoh bu den HomeriBCheii GedlahtoQ. Fur l 

Schiilgebraueh hearbeitel. von Georg Autenvit 
yielen Holz&cUiutteii und drci Kart.en. Dritte iuDgearbei 
Aullage. gr. 8. 1881. geh. 3 A 

Worterbnoh zq Xeuophans AnabaBis. Von F. VoUbreoht. ' 
iOt 75 HolKachnitten, .3 lith. Tafeln mid 1 Kart.e, 5. Aull, 
gr. 8. 1882. geh. 1 ^ 80 A. 

BchulwBrterbuoh zu O. J. Casar mit besonderer Bertiok* 
sichtigiing der Phi-aseoiogie von Dr. U. Eheling. Zwdit« 
Aiiflage, bearbeitet von Dr. A, Draeger, Direktor da» 
GymnasimuB au Aiuich. gr. 8. geh. 1 ^ 

aDBrltrbui^ ju ben StlienBieft^tcibuiigcn bte eorndiue ntpDi. 

^iit beit ©t^ulgefirauc^ (lerouSgeiifbcn udu ^, §aaii<. 

7. Slufl. 8. 1882. flel). 1 jK SPiit beiii Sctte bc3 *JfcpoS 

1 Ji -20 \. 
ffiortrtbu^ ju OoibB aRelamorti^Dfcn. eon 3. ®tt(cli>. 

Sritte Slnftage, bejorgl uon Sr. $dUc. gr. 8. 1879. 

gc^. 2 J(. 10 \. 
ffiBortetfiu^ ju ben gabtln btfi ^^iiQruS. ^iir ben &^uU 

gebtauc^ ^erousgegetien Don 1. ©i^auljnctl. 2, 3[nfl, 8. 

1877, ge^. GO 5^. aKil bem lejte be§ $^dbru# 90 \. 
WBrterbacb zv. Siebelia' tirooinluiii poeticum. Von 

A. Schaubach. 6, Aufl. gr. 8. 1882. geh. 45 \. 



OBUM (iRjVECORUlI KT EOllANOBim 1 
LTEUBNERIANA. 



I 



EITCLTDTS 

RA OMNTA. 



KIHUlillHNT 

IIBGRO KT H. MENGB. 



rCLIDTS EIjEMENTA. 



I. I.. HEIBGRG, 



& n. ^jBBoe V— uc v 



m 



LIPStAE 



fiTBLT 

SCBIPTORnJl GRAKCOR0J1 I 



l 

^H. lcscliliies ed. Fraitkt. . . - 
I iMckylu «i- Dmdor/ . . . 

«[nislne HtackB . 

llbertl Tr«llBa ed. Umdnrf 
imnilaHita IlBrti. ei. Oardtii 






kndaciSt» «d. Slap. £d, n . 



InttalinL 

(DthDlOE 



i Uttnft 



- Irrro ed. Berai. Kd. I 
mtlphon ed. 8la/i. £d. U 
AntnnlDna ri. filicA . . . . 
ApolJodorBi ed. SrUa*. . , 
AppDllon. Bhodlog ed. Ktrt 
ippian sd. JfmiclMoAH. i t 
A[Ohiini>diB Dperl amnU. 

irUtDpliRiieB ed. Btrsli. 3 ■ 



^ 



IrlslDteltt de purtlbi» ■nlmai. 

phrslca ed. FraHil 

GthiuNlcDmnebeled.SiunnMI 

£thic* Endemla cd. Si^nmm 

de eoelo etc ed. JVanM . . . 

de colorlbns, andlblilbug, 

nhrilognoinoiili» ed. Franll . 
palitlci -■ " 






mlU 



rllll ed, flis*l . 

A(rl«nl eipedltlo ed. Abidil . . . 

eerlpta nln. ad. BerclieT . . 

AthDDHen* ed. JUei-uki. t idU. . . . 
Angnitinnaltet.ed.ZWlAan. StoII. 
■ — ■--■» ed. /Wper 



ArUDHS 



- de eoRiDlotlone 



- da bello flklllco. Ed. DiiiL. - 

flitlli. Kd. miu. . . . . ■ 

GmiiilBti Felli ed. Rou 

VKtull» Bd. Iliilltr 

CMmIId), Tibnllni, Properllns . 
.-.-lil UIidU gd. Hroiihn . . . , ■ 

L Cetne »d. Daremliara 

P Cvutorliin» ed. HiOiteii ...... 

Uleera ed. UiUrr. Fin n. Tol. I, 
Pui IV. Tol. I. U. in, Jedei I. 
_ ed. *1«i. 5p»rt. lliDU,Vpll, 1 
— ' oral. lelectM el XloSi. t pui, 
oratlanH ael. edd. Eirrliiani 



THECA 

ROMANORUM TEUBHERUNA. 

€lc«ro,«pUtoUeBelect«eed,i)iiiiicl. 

epUtoIae »i.Wtteiiberg. itoU. 

ConimodUnai ed. hudteis. I A 11 
Camelins Mepoii ed. Uaiyn .... - 
Comntnii ed. ijum . - 

PhrrKlnt ed. Meiner. . , . 
i tdU. 



Phrj^gint ed. Ji 

nemoittbeiieB ed. Siim 

Anoh fn 6 Pertei, i 

Dlelji CreteniU ed. Mr!iirr . 

Dlnftrchni ed. Blaji 

nio Ceuliii ed, dindorf. b ti 
DId Chryioat. ed. DindorJ. : 



^eibmi, L 






EndoeUe ilolitriniB ed. Flaili . . 
Eiirlpide&ed^auci.Ed.in.TaUAIIi 

Tol.m, FrBgffienl» 

— Einieln ~- " ■ 

euKebinB ed. OindorJ, 
EuIroplBB ed. Dielacli. 
Fabnile AenoplMe ed 
FabnlBeHonen.ed.f^b 
FlDrni ed. Balm , . , 
FronlinuB ed. Brda-idi 
«■ini Bd. SBMlAii . . 

Hellodor ed. Bttier . 
" ■ ed. ifebter . 



Herodotna ed:. Dleu^ S ti 
HeBTDblM nileilM ed. FlaiA, , 



Itatoriot (Iraeel mlnoreg ed. fKn. 

dorf s Toii. 

liatDrtEOmra Itoni. reU. ed. FtUr. 
loueri UUb, kplt. i Bcnd mii 
ElDleltusg TOD &ii^i 

"' '-It. 1 B 

I Snffeb 

. I (IlllB 

— — 1. B (UiBt o) . 



HrslBl Oromitlei llber ed. OemuU 
Hirnai ilomerlct ed. Baumeluer. . 
Hfperides ed. Bla/i. £d. U . . . 
IlUdli e^rml» ed. Ktie-Alf. . . . 
iBDerti kutoriilib.deCr-- -- 



JOBepbaD,PUtIna,ed. BiUar. e ' 



JOBepbaD, PUtlna, ei 
liMu ed. BditOe . 



i iin 



EUCLIDIS 



OPERA OMNIA. 



EDIDERUNT 



I. L. HEIBERG ET H. MENGE. 




LIPSIAE 

IN AEDIBUS B. G. TEUBNERI. 
MDCCCLXXXIV. 



EUCLIDI8 



LE ME N T A. 



EDIDIT ET LATINB INTERPRETATUS EST 

L L. HEIBERQ, 

DB. PHlXi. 



UOL. 11. 

LIBROS V— IX CONTLNENS. 




LIPSIAE 

IN AEDIBUS B. G. TEUBNERI. 
MDCOCLXXXIV. 




IiIPSIJLB: ttpxs b. o. teubnkrt. 



PRAEFATIO. 



iis Elemeniornm libris^ qui hoc continentur 
le, emendandis pro fundamento habai codices 
, de qoibus uideatur breuis^ quam dedi uol. I 
— IX, notitia; codicem Bodleianum B in libris 
IX ^) contulit H. Menge. Parisino 2466 (p) 

libro YII uti potui, neque magni est momenti. 
m omnium Theoninorum optimus codex Lau- 
us F inde a VH, 12 p. 216, 20 ad IX, 15 

6 deficeret — nam eam codiois partem, quam 

fp significaui, prorsus inutilem esse, adparet, 

i re in prolegomenis uoluminis IV uberius 

— , et cum cod. Bononiensis b (u. uol. I p. IX) 

3ntino in hac quidem parte non longe distaret, 

Vn, 13 ad IX, 15 hoc anno Bononiae contuli 

loco scripturae discrepantiam notabo. ad sup- 

m adparatum criticum in libris VIII— IX etiam 

arisin. Gr. 2344 (q) membran. saec XU contuli, 

Hauniam transmitteretur, intercedente prae- 

bibliothecae regiae Hauniensis a liberali- 

bliothecarii Parisiensis LeopoldiDelisle facile 



In his duobns libris ab VIII, 17 de v littera, quam 
ninov uocant, uel omiBsa uel addiia in B nihil in 
ae adnotatam erat. 







VI 


TRAEFATIO. 


impetri 


lui. huiuB codicis scripturas inde a p. 373, 15 


auia locis in adparatum recepi, reliquas ab initio ' 


libri VTU hic dabo. j 


p. 216, 


24: (5ff< b. 


. P- 218, 


9r xk avza\ om. b. 




18: iv} «tti h b. 




27: fWv] om. b. 1 


p. 220, 


1: -cov Z] Z b. 




11: ij] uidetur eras. h. ■ 




26: Sataij htiv b. ^^^^J 


p. 222, 


2: ^ov^Oi] YOVfievot b. ^^^^^^H 




7: ^j corr. es 6 m. 1 b. '*^^^^^l 




14: A] corr. ex ^ m. 1 b. ^^^^M 


p. 224, 


^^^M 




24: TtolXtatkaaiasaai b. i^^^^^l 


p. 226, 


xaQ ^^^^^H 




6: WEiioVt b. "'^^^^H 




17: UQt&tLOi] a^a a^i&fiol h. '^H 




25: menolijxe b. ^^| 


p. 228, 


2: (JU' us] to; 6i b. ,]■ 




6: 7C£R0/l2M b. ^M 




21: sequitur p. 428, 23—430, 17 b {«'}. *^l 




p. 430, 11: iaxlv] om. b. ;S 




13: Jicd] h b.>) M 




16: OTCE^ ¥dci dciiai] om. b. |^| 


p. 230, 


16: eZ] sapra scr. m. 1 b. ^M 




&ot litf/i'] punctia del. m. 2 b. ^^H 




a^i9fioi] lOoi b. ^l 




^i;ii];io(;] aiiqloi; cfff^i' b. '•H 


p. 232, 


2: ior/v] om. b. H 




4: EZ] EZ &Qa b. .^1 




7: Bequitur p. 430, 19—432, 8 b. (k|9'). ^I 




p. 432, 7: iazi] om. b. ^l 




8: »y' b («' edii. = xa' cod.). ^M 


1) SecipieDdum esL ^^H 



FRAEFATIO. YII 

9: iJJ/i^iovg] noXlavg b. 
11: aU^Xovff] noXXovg b. 
14: fM^] i^i ilOiv ot A^ B ihi%unoi tmv tov av- 

f^ Xiyov i%6vtmv avtoig b. 
18: imQOvOi} bjIL fie- in ras* m. 1 b. 
20: f&v ^] in ras. m. 1 b. 

8: totg] tm b. 
11: jcd' b et sio deinceps. 
17: ilot] iloiv ot AjB h. 
18: mftovg] tovg A^ B h, 
21: lotmoav] litt. or corr. ex 17 m. 1 b. 

1: itwohixe b. 
12: motv] iUsiv comp. b. 

3: moi b. 
12: ante tig est — in b. post A^ E uacat linea in b. 
13: i^] ii h. 

22: A, E TtQmoij ot ii] om. b, in extrema pag. 
26: Tov] nqog tov b. 
, 1: Tov] TT^g Tov b. 
2: post E est — in b. 
JB, r] r, B b. 
24: &<si, b. 
(, 4: Tov] To b. 
8: dtf] « b. 

E, z/] z/, E h}) 
16: a>tf» b. 
l, 3: E] in ras. ra. 1 b. 

22: cotft b. 
6, 9: r/4] ^r b. 
8, 1: ft»i] supra scr. m. rec. b. 

14: fl^T^^] flBtQEt b. 

0, 1: 6 B] To B b. 

6: 7}yov(i€vov] corr. ex fiyov(jLevog m. 1 b. 
9: sequitur p. 432, 10 — 20 b. 

p. 432, 10: SXXoDg xo X§' to i^rjg b. 

I Hoc ergo ex P recipiendum erat. 



ym 


" 


PRAEFATIO. ^^^^^^1 








p. 432, 13: sOTw] iffrw 6. b. ^^^^^ 








19: B\ eorr. ex f m. 1 b. ' 








20: ioit] comp. b. 








Sjiep i'dH rf*t|a(] eomp. b. 


p 


260 


10 


A}'' b et Bic deiuceptj. 






17 


yij-ovofi Sv eZr) to ijetiox^ei'] rf^iou Sv itrj zo 
tiijoviuvov b; item lin. 21. _ , 






24 


Bi] Tov npo («iJTOw, os xal tov A (wipjjOEi. i? b- 


p 


252 


1 


hiQov] Tov hiQov h. 






13 


ixi.xax^&iv'] S^t^ovfxevov b, mg, m. 1: y^. xi 
iitayya^u. 






19 


xovg nwouj Xoyovs b; item lin. 22 — 23. 


p 


256 


21 


(i.iiQovat. b. 






25- 


5] xoi 5 b. 


p- 


258 


8 


post inonevog reliqoa pare lineae quasi oma 
mentie quibDEtJam expleta est in h. 






9: 


Tovg] lov b. 






13 


roO r] Tov r, OTttv ol A, B ■n^&xot ji^Aj 
allrikovq ataiv b. 






20; 


HetQovoi b. 






24: 


faioHrai'] hoviai b. 






26: 


iZ] e corr. m. 1 b. 


p- 


260 


4 


SQa]Jea (Oi b.') ( 






16: 


fiezQaatv] (liT^MjowOi b. 






25: 


^^ijoovffi b. ^^^^^^t 


p- 


262, 


11 


Si I^^^H 






13: 


ftrt^oiiot ' ^^^^^^1 






14- 


nex^aovat ^^^^^^^^1 






16- 


fitiQovOi b; item lin. 17. ^^^^^^| 






23: 


fiEipovOiv] linp^tsovai b, J^^^^^^| 






24: 


r] iu ras. m. 1 ^^^^^H 


p- 


264, 


3- 


HezQovOi b; item lin. 4, 7, 8. ^H 






13: 


lov Z — 14; fuxQovfievos] om. b. ^| 


p- 


266, 


10. 


To «^Td — 11: «^'^fo"] om. b. ■ 


p- 


268, 


9: 


ino] h i7t6 b. ■ 




1)1 


. 26 


^AiM 





PRAEFATIO. IX 

11: 6 H aQa] intl b H %m6 tmv z/, £, Z fu- 

fj^irai, i H h. 
.4: jiii}] fM} i H iXaxtCTog cSv IxH xit Ay JB^ F 

L7: filfftfi b. 
19: «Sv] om. b. 

VIII. 

L8: xm — 14: nkrfiii\ oin. bq. 
18: ^^a»v«— 19: o tc] om. bq. 
12: xiccaqii^ J b. 

20: lirriv] «^fiig 4^ 6 ^ tfvo tovq A^ B nok- 
hxnlactacag tovg Fj A 7ttnolfi%ev' lcttv bq. 
20: &Qa] om. b. 
21: fiiv] om. bq. 
2: 6 r] ovtag r bq. 
3: 6 A] ovtmg o A bq. 
4: WilXxiCwCag b. 
8: 6 Z] ovToog 6 Z bq. 
10: 6 If] ovto)^ 6 If bq. 
11: h A\ ovtmg 6 A bq. 
15: aAA'] ideCx^rj dh %aC bq. 
23: eicC q. 

of -4, B — 24: eiolv] snpra Bcr. m. 1 q (slcl). 
26: dl Tcov] di tov bq. 
3: tor^] corr. ex avtotg m. 1 q. 
9: tiaaaQBg] d q. 
11: iav] supra scr. m. 1 b. 
1 : %al insC — 3 : lavtov fiiv] ot cr^a ttx^oi av- 
tmv ot A^ S ngmot nQog aXXi^Xovg elaiv, 
inel yccQ ot E^ Z nQmoi^ i%axeQog dl av- 
tc3v iavtov bq. 
6: %al] om. bq. 

%a\ ot — 7 : elcCv] nQmoi %al ot A^ S bq. 
14: eiciv] inei bq. 

aXXriXovg] aXXriXovg eialv^ iaog Sl 6 fiiv A 
TGo Ay 6 dh S tm A bq. 



X PRAEFATIO. 

p. 278, 18: avdXoyov] om. b. 

22: Z] in ras. m. 1 b. 
23: avaXoyov] om. bq. 
p. 280, 1: Kccl] om. bq. 

6: 0] e corr. m. 1 b. 
10: 0, H] H, e b. 

avccXoyov] om. bq. 
11: Kal iv] Kal iv rs bq. 
13: 0, H] H, & bq. 
14: avaXoyov] om. b. 
15: iv To5] Iw bq. 

16: Xoyoig] Xoyoigj Scovtal rivsg t(5v H, 0, X, -^i 
iXaCdoveg aQi^fiol IV T£ Torg TOt) A TtQog 
rov B xal tov F TtQog rov A %ai hi rov 
E TtQog rov Z Xoyoig q. 
17: ovrcog] om. bq. 
20: iXdacmv] iXdrrcDv b. 

iXdcaova] iXdrrova bq. 
21: T6] om. bq. 
p. 282, 1: J3, r] r, B bq. 
2: (lerQOvCt bq. 
Twv] TOV q. 
4: 6 H] (prius) supra scr. m. 1 b. 
6: 0, H] H, bq. 
8: Tov Z] Z q. 
9: vTTo] liTro bq. 
12: 0, H] H, bq. 
14: insl] xal ijrf/ bq. 
20: Iddxig] ocdxig q. 
22: avaAoyoi/] om. bq. 
iv] IV T£ b. 
rs] om. b. 
23: irt] om. bq. 

24: Iv] el yicQ fwf «tetv of iV, S, M, O l^ijg 
iXd%i>(Sroi bq. 
p. 284, 1: el yccQ fttj] om. bq. 
2: avdXoyov] om. bq. 



FRAEFATIO. XI 

6: oSn»(] big q. 

7: «•] om. bq. 

0: ^MQov6i bq.; item lin. 15. 

0: ivaloyw] om. bq. 

1: foy] om. bq. 

3: fov] (bis) om. bq. 

S: a^] om. b. 

ovaloyoy] om. bq. 

LO: r, £, ^] in ra& m. 1 b. 

L6: sco/] om. bq.^) 
i6: nmohfHtv] (prios) mnoltfM q. 
L7: A] 6 oorr. m. rec. b. 
18: A] 6 corr. m. rec b. 
ag ii — tov S] om. b. 

7: fftcreg] f»«t^r q. 
18: ftHQOvCiv] (ii%iffjcovCi bq. 
14: «2 — 16: viv i^ Uya yig oxt ov (utqsi b A 

•tov r bq. 
16: xal ocot] ocoi yag bq. 
17: To^ A] in ras. m. 1 b. 

1: 1}] <2 q. 

yao] yccQ Z q. 

6: fitCQi^cei,] fUXQ€t bq. 

9: (UtQy] (UXQet q. 
14: ov] ftij q. 

ovdi] ovd* q. 
15: (UXQrjcei] (utqtiCh' OTttQ icxlv SxonoV wto- 

xeixai yccQ 6 A xov A (UXQBtv q. 
16: 6] xo q. 
20: (uxa^v — avaAoyov] om. bq. 

8: r, J, B] B, r, J bq. 
10: elcl q. 
11: elcl q. 
14: %al — 15: tov Z] om. q. 



itaqae quoniam bq p. 286, 13 sq. cam P consentiant, 
rheonis in adnotatione ad locnm illam toUendam est. 





^^^^ 


Xn PRAEFATIO. 




p. 292, 18: ixovtag] i'xovxug avTOig bq. 




22 1 mC] x((i q. 




p. 294, 1: £10/ q. 




xol oJ: — 2 : cielv] om. b. 




3: «H om. b. 




10: «0( bq. 




14; nsTa^v] f|^s ftEWf^i; bq. 




19: luta^] aupra acr. m. 1 b. 




20: iiiTCcmcoxaGiv] ifatlittoviSi.v b. 




21: r^sj i^s £ bq. 




p. 296, 1: ^CTtol7\M bq; item lin. 2, 3, 4. 




6: Z, ff] H, Z bq; item lin, 7. 




p. 296, 10: rwi'] om. b. 




iottv o] Eflti Xttl 6 bq. 




12: Kgn io'v] apa to q. 




fisipcf] om. b. 




p. a98, 2 


roos — 3; A^oSiMtiiA ie 


n. f»OS bq. 


G 


H] K, ut uidetur, q. 




8 


ToatMJiot] oiliMs b. 




12 


i'] om. q. 




iKaTceow] om. bq; yp. sxon^pou m 


g. m. rec. b. 


15: (letaiv] l|% fiEioigi; bq. 




21: 0?!.] cori-. ex Sk q,') 




p. 300, 8: a^a] om. b. 




10: m^olrixs bq. 




11: E] e corr. m, reo, b. 




13: di] om. q. 




15: E] corr. es 9 m. rec. b. 




16: n£icol-i]ii£ bq; deinde add. b mg. 


m. rec; rov 




mo/ijxE. 


^iv] om, b. 


J 


17 


7t(noh}»e bq; item lin. 18, 19. 




19 


^iv] om. bq. 


■ 


23 


ital us — 24 : 101- H] snpra scr 


I». 1 q. ■ 


25 


r^v] t6v q. 


j 


1) P. «0 


8, 21 ia adBOt. addfttur: «] om. BV 


^^^^ 




i^H 



PRASFATIO. Xm 

f: iUi^ ig i E itQog tov] in ras. m. 1 q. 
I: tmf] Tov q. 
): JC] in ras. q. 
A] in ras. q. 
): B] e oorr. zn. 1 b. 
2: xcrl o&c--* 18: tov A] om. bq. 
1: r yuQ] yiiQ F bq« 
ft: mnohpu bq. 

8: #ia nr «vra di^ «a/] jraJUv ^l j F tov J 
%ollemlttCw6ag tov E jtsnolfimv^ o 6h J 
{avTOV 9CoiUla7rXaaiaaaff tov JB 9re9ro/i}x<, dvo 
d^ aQi^fiol ot r, J Sva xal tov avTOv tov 
(om. b) J noXlanXaCuic^fftig toifg £, B 
mjtoiiqnaciv' Sattv Sga bq. 
9: JB] JB. aXV mg 6 F ngog tov ^, ovtmg 6 

^ 9r^og Tov E bq. 
0: a^] om. q. 
1: a^id|[io^] a^iO^^ 6 J£ bq. 
2: iavTov] Iovtov (Uv bq. 
4: Tiov] Gorr. ex tov m. 1 q. 
6: %al i F iavtbv noXXanXaCidcag tov E m- 

nolfjnev] om. bq. 
7: lUv] om. bq.^) 

nenoCriKs bq; item lin. 8. 
LO: ns7tolri%s q; item lin. 11. 
27: J] Jy ovtag ts (om. q) 6 JC nqog tov B' 
ideix&ri di xal tag 6 F TtQog tov J bq. 
Te] T€ 6 bq. 
4: Tov] om. q. 
8: TfS] om. q. 
10: fiiv 6] 6 fiUv bq. 
14: TeT^aycovog jr^off TeT^aywvov] TCT^ayovog aQid'- 

(wg nQog tsxQaymvov aQt^fWv bq. 
22: e^0»v] comp. ictiv corr. ex comp. eictv b. 



. 806, 6 in adnot. scribatnr: „6. xal 6 — sreiro^i^xcv] 
'heon (BVep). 7. fiiv] om. BVqp." 



^^^ 




XIY 


PRAEFATIO. ' 


p, 310 


23: B] e corr. m. 1 b. 


p. 312 


1: daiv] eiai bq. 




4: naXiv — netQihco] aki.o iJ^ (UtQsirio o I to»i 




J bq. 




7 


BJ in ras. m. 1 b. || 




10 


^, E] in ras. m. 1 b. | 




15 


ojiEp sdft ^Eilat] otQ. bq. 




18 


xai iav—20: nsx^jjati] om. b. 




25 


h Si J - 26: rii' ^] «al fw 6 P rov ^ 




TCoHcaiXoaiaaae tov Z naiebio, b di J iav- 




t6v bq. 




20: Z] H bq.' 1 


p. 314 


5: £i9 q. 




10: d^'] om. bq. 1 




11: o/] xai o/' bq. 




12: TCQog xov] wpdg bc[.') 




13: ws] supra scr. m. 1 b. 




22: i^i»f*ol] om. bq. 




24: ^iTQsi] fiETQTjau b. 




26: ti yaq nBxgci b F cov J, ftErpjJaei] mg. m. 




rec. b; cl yuQ b F zbv d h%xqh, ,i«Tpr'eEi q. 




26: oiiiJi] oiS' bq. 


p. 316 


3: yap] yie fi.; b, aed fi»; eras. 




««^] corr. m. rec. b. 




5: ojcEp ?*« d«loi] om. bq. 




21: o««9 «£. det^ai] am. bq. 


p. 318 


1: 0^010»] om. q. 




13: n^oAvi^iUi 01(10«? b, Bed ayll. Au in raa. m. 1; 




item lin. 15, 17, 18.') 




14: ntnoirjxs bq; item lin. 17, 23, 




17: A] corr. es H m. ree. q. 




22: nolwtXaataaag b; item lin. 23. 




28; ei^Oi q. 


p. 320, 


4 


fS%] ^l aez'7s q. 


l) Etgo 


Tov cam P omittendam. 


2) Itaqae fortaase haec forma nocabuli in hac prop. ODm 


P sernttnda eat. 


^ 





PRAEFAHO. XV. 

& rj 816 bq.*) 

4] md b. 

ol] iQt^ftol of bq. 

£] E iQt^fiol q. 

cnQiot} evsQBol oQt^itol b. 

(tiv i] sio bq.^ 

fud] ^ bq. 

ywrf *? q. 

T&v J] sio bq.') 
L: «20/ q. 
h iui(] Sfriv j^ mg K itQog %ov M^ 6 M 

ftifog %ov A^ nai q. 
7: mfgolfpa bq; item lin. 23, 25. 
3: Jlf, ^] ^, M bq. 

4: Jicr Tcir ovra d^ xcr/] TsaiUv Ittc/ icvtv mg b 
J n(^g xov £, ovxmg 6 J7 JtQog tov 6, 
^iUaf a^ iailv bq. 
6: M, ^] ^, M bq. 

efoiv] om. b. 
9: N] corr. ex H m. rec. b. 
1: r, 4 E] 4 £ q. 
14: A] con\ ex J m. rec. b, 

10 v] Tov ix Twv Z, IZ" Tov bq. 
17: N] corr. ex H m. rec. b. 
IS: xov] om. bq. 

Tov] om. b. 





N] 


corr. ex H m. rec. b. 


JO: 


K 


e corr. m. rec. b. 




xa 


. G}g] 6g bq. 


1: 


z] 


in ras. m. 1 b. 


5: 


iV' 


corr. ex H m. rec. b. 


6: 


/y 


H xal 6 E TtQog tov S q. 


9: 


iV' 


corr. ex H m. rec. b. 



adn. p. 320, 8 delendum ,.corr. ed. Basil.". 

adn. p. 320, 19 deleatar „o fiiv Vcp**; habent fihv 6. 

adn*. p. 320, 26 addatur: „26. tov J]x6v fihv JBYtp,'^ 











XVI 


PKAEFATIO. 




p. 324 


11 1 rov] bis b. 
12: ff] E q. Bj 9 q. 
13: ««;] «bI tSs b. 
26: ^U' (is] «s d^ b. 
28: «H om. bq. 




p. 326 


7: o£] om. bq. 
10: aeiftfxos r] r a^t&iAOg bq. 
13: ^, r] ^, B, r mutat. in A, F, B m. rec. b; 
^, r, B q. 
E] Beq. fffttv oe« we 6 ^ re^oe tov E. A 
n^S -cov r. oU' mg b A nQig tov T, 
DVTOJS r (corr. ex ^ b) npog ibv B. 
KKt ag cfptt .i/ npos "v E, o F ngbg 
Tov B q et mg. m. rec. b. 
iiSaKtg] mut. in aaaxtg m. rec. b. 
apa] mutat. in 6i m. rec. b. 
14: K«l E — 15: furpff] om. b. 

6,n Si q.'} 
16: nEnoti}xi q. Seq. tw di E TtoUanXaaiaaag 

Tov jT jicwo/ijKEi' q et mg. m. rec, b. 
17: ian q. of] af q. 
19: r, B] B, r bq. 




p. 328 


3: Z — TOT A] jxctwpos toIv Z, H toi. £ 

b: A] Z bq. Tiv £] Hbq. 
6: ^-5] om, bq. 
rov] om. bq. 

nolir — 9: lov B] om. bq, 
9: t6v] om. bq. 
10: ToV] (priuB) om. bq. 
11: ToV] om. b. 

xtt/~12: lov H] om. bq. 
13: opi&fto/ slaiv] eiOiV apt&fioi bq.') 






17: oii.oioi] om. b. 


1) In adn. p. 326, 14 aadatur: „14. ff^l corr. ej Be S", 




in adn. 


ad p. .126. 20 deleatnr „et B (corr. m. D". 













PRAEFATIO. XVH 

23: /1] Ay fi bq. R\ H, S b, sed corr. 

26: sM q. 

26: 6 Z — iifi^itol] om. bq. 

2: xav JtQo] om. bq. 

4: Tov] om. bq. 

5: Tov] om. bq. 

%al] supra scr. m. ree. b. 

6: toig] xoi b. 

%al—7: Ay T, A\ om. bc^. 
12: o %{\ oxi q. 
17t N] corr. ek £r m. rec. b. 
18: 9MsfeloA|M bq. 
36: N] eorr. ex IT m. rec. bq. 
22: 6^] ii bq. 

£] H bq.O 

1: r] B bqO 

6: mTeolfiM q. 

6: i(»/v] om. b. doiv] om. bq. 

7: slci q. 

8: Tov] corr. ex to m. rec. b. 
12: xov M] M q. 
15: S] post ras. 1 litt b. 
16: ofioioi] ot q, om. b. 
19: xglxog] y b. 
22: Xiym] Xiym dff b. 
24: r] e corr. m. rec. b. 
25: ttat q. 

26: -x^yfovoq di o A re-] mg. m. rec. b. 
r] B bq. 

7: icxlv] laxcii bq. 
12: %d'] om. q. 
14: ovj corr. ex ^ m. rec. b. 
15: XBTQciymvog fi] ^ xtx^ymvog bq. 
17: poat B infl. Jlo^rov m. rec. b. 
Xoyov] om. bq. 

Q adn. p. 380, 22 addatur: „6 £ tov T] 6 H xov B 

Idei, edd. Heibcrg ct Monge. II. b 



XVm PRAEFATIO. 

p. 334, 19: laxcai] idxM q. 
22: sl6i q. 
23: r] in ras. m, 1 b. 

rov] om. bq. 
24: tov] om. bq. 
p. 336, 8: ^] e corr. m. rec. b. 

dij] di b; om. q.^) 
10: yciQ of] yciQ o b. 

ofwtoi] Sq€c ofioioi bq. 
11: slat q. 

12: (ura^v] in hoc uocabulo desinit q fol. 165^; 
XelTt, g>vXXa 'rj mg,\ rursus incipit p. 372, 15, 
n. n. adn. {ivtavd^a Xslnovdi qyvUia ^ mg. 
fol. 166'). 
p. 338, 5: tSTQaycDvoi] tsxaQayiiivoi b. 
22: E] e corr. m. rec. b. 
25: oTtsQ idei det^ai] om. b. 

IX. 

p. 340, 9: A] e corr. m. rec. b. 
10: 7te7tolfi%e b. 
14: di] om. b. 

17: rwv] corr. ex tov m. rec. b. 
19: oTtSQ idet dei^ai] om. b. 
p. 342, 4: aQid^fiol] om. b. 

5 : ectoadav — 6 : 7toteha>] dvo yccQ aQiS^fiol ot 

A, B TtQog (mutat. in TtoXXaTtXaCidiSavteg 

m. rec.) aXXi^Xovg tetQaycovov tov F TtoieC- 

tcDiSav b. 
11: liStiv aQa] om. b. 
12: Tovj bis om. b. 
14: iftithtxei] iftitlmei aQiS^fwg b. 
17: idv — iimhtti[i] om. b; &v de dQtS^iiciv elg 

liicog dvdXoyov ifiTthtcei mg. m. rec. 
18: of aQa] aQa ot b. 



1) Itaque dij cum P delendum, ut suspicatus eram. 



FBAEFATIO. XH 

9(M0M|M »• 

n^ f^] ngog b. 

tiv 2^] 2^ b. 

foS A] onL b; post a^i^|M)v ins. m. rec. 

tov] om. b. 

iimtmattiv] inMmthiaCav b. 

dcvr^g] tkoqtog b. 

far/v] om. b. 

Qfu\ om. b. 

yv^ .^] A yi^ b. 

o( ^, B] ante ras. 2 liti b. 

A] corr. ez ^ m. 1 b. 

jsv^ B(fa iitvl] Itrriv cr^ b. 
h ^] ftifmtog b. 
L: sMffo^^ b. 
i: lovrov] lcrvrov ikiv b. 
L: 6 ^ — 22: tov B] xov i\ B itolXtatXa^ii^aq 

xhv r nsKohjpniv b. 
): %ttl &g\ mg b. 

i: i A] ovxmg 6 A h, A e corr. m. rec. 
3: i<sxi xipog] ioxt 6 %vpog b, sed 6 deletum. 
1: wto] corr. ex wtiQ m. rec. b. 
4: ijul — 15: (lovadag] om. b. 
5: Tteitohixs b. 
7: 6 ix] Ix b. 
4: itsxai] iiSxl b. 

o] TCcrvtf^, 6 b. 
1: Ttavxeg] om. b. 
2: post dtaXelnovxsg add. Trcrvre^ b. 
4: oTi] om. b. 
6: Ttavxeg] om. b. 
8: a(ia] aqa b. 

Ante xsxQayoDvog eras. 6 b. 
9: jrcrvT^ff] anavxeg b. 
10: Post ii ras. 1 litt. b. 
12: fiovcr^] 17 fiovog b. 

A^i^&iuov] om. b. 

b* 



XX PRAEFATIO. 

p. 352, 14: rcS A] avx^ b. 
15: TtETtolriKe b. 

17: Kal b A aqa\ aqa xai 6 J h, 
20: ^vrsg] om. b. 

xitaQzog] A b. 
23: A^ A a^i%\>.ov b. 

otfrooff — 24: a^i,%^6v\ mg. m. rec. b. 
p. 354, 3: 7te7toiri%e b; item lin. 4. 
7: o] m. rec. b. 
8: iiovadog] lAOvddog & Z h}) 
12: fMvddog] rrlg (lovddog b. 

^k^S — 13: aQi^fwi] aQid^fiol l^ijg b. 
17: (lovddog] rrjg jnovddog b. 

p. 356, 10: tira^og] A b. 
15: B] JB [lexQet b. 
21: €^(y* b. 
p. 358, 8: fiovddog] trjg fiovdSog b. 

ocoidfiJtotovv] o7to0oiSri7toxovv b. 
22: OfMUog — 23: hxi] om. b. 
25: 6r[\ om. b. 

fcrro) 6-4] corr. ex eiSxaCav m* rec. b. 
oid'] oide b. 
p. 360, 5: Toi/] bis om. b. 

16: rfra^rov] A b. 

19: |[*ova^og] xrig (lovdSog b. 

20: IXatfcrcov b. 

23: [lovddog] xrjg fiovddog b. 

25: ildxi6xog\ ildaamv b. 
p. 362, 8: TtoQidfia — 11: avrov] om. b. 

17: OTtoiSotdriTtoxovv] o60L6ri7toxovv b. 

22: (iri yd^] [i'^ yciQ (lexQetxo) o E xov A b. 
p. 364, 1: £] corr* ex -4 m. 1 b. 
3: iiexQelxo)] iiexQelxm di b. 
4: %e7toiri%e b. 

1) In adnotatione p. 354, 8 addatur; „fioWdop] (lovdSog 
6 Z Theon (BV9)**. 



FBAEFATIO. XXI 

29: Ijfivtug] ixwvag avtoig b. 

8: ^yovfuvov] xov ^yovfuvov b. 

6: v9ro] uQi^fMA v«6 b. 

7: ov] om. b. 
L4: l|^d om. b. 

6: nag] aiuig b. 

6: i E — 7: (UtQeixai\ om. b. 
12: i Z ovx iati] ov% iativ o Z b. 
33: $1 yao] si yif^ l<m HQmog b. 

2: wtag — 3: iistqmui] om. b. 

3: 6 Z a^ wto nqmov] vno ytQmov aqa b. 
21: avalioYOv] aloyov b. 

1: V7t6] i% x&v b. 

6: ^] e corr. m. reo. b. 

7: OJUQ idu dBi^ai] om. b. 
20: mnoCfjKB b. 
22: noXv7^aui(S€ivtsg b. 
23: tov] corr. ex aitov b. 
25: iutQi^<sov(Si b. 

2: fUtQoviSiv] iisxQi^<Sov<Siv b. 
14: OTTOMitoilv] onoMvv b.*) 
20: TcenolrjKS b; item lin. 21, 22. 
22: slai b. 24: eotfi b. 

2: hxi b. 

3 : iciv di — 6 : SiSxe] %al b. 

6: Z^] ^Z b, sed Z e corr. m. 1. 

6: ^E] ^E aQa b. 

5<fT6 — 7 : i^iv] om. b. 

8: yaQ] di b. ix] otco b. 
10: itfnv] l<Triv. a[(»£ 6 ix toov Z^, ^£ %al 

nQog xov ano xov EZ TtQmog iiSxiv b. 
13: i<sxw] i<sxi b. 
17: ei<Si. b. 

19: %al] &<Sx8 %al b. i%] vno b. 
21: i%] sic b.*) 

Q adn. p. 374, 13 Bcribatur y^izovtmv Xoyop V". 
irgo in adn. p. 376, 21 nomen Theonis deleator. 



"Opoi. 

a'. Mspos iorl (iiyB&og fieyi9ovs lo llaeeov xoi 
fiti^ovos, orav xatafuxQtj to {ift^ov. 

/J'. nokXaxXaaiov dt to (ibi^ov tov ildn 
6 0T«ii xaTaii6T(fiJTai vno tov iiditovos. 

y'. Aoyos ioti Svo [isyt&iav ofioysvav j] 
aijlixorijtd ^oia gx^ols- 

6'. Aoyov 'ixsiv rcpos aXlfika fteyi&rj Xiyerttt^ 
a dvvatai itoi.i.axXa(Sia^6fi£Vtt dXl-^Xav v7iaqi%Etv. 
10 t'. 'Ev za avta Xoya (tsyd&tj Xiystat- clvat 
nffiSrov Jtpog SevTepov xal rpitov ;rpoe titaQtov, 
otav ta tov TtQtatov xal tQitov lCaxLs 'XoXXaiiXdonc 
rmv loiJ Sivriffov xal tstdfftov iadxig aoXXajtXaeiav 
xa&' oaoiovovv %oXXanXttaia6fidv ixdrtQov ixariQm 
16 ^ a^a vnsQijjri ^ KfiK Hea r^ ij afia iXXsinji Aij^id-^VT 
xaraXXi^Xa, 

c'. Ta Si roi' avrov ix^vza Xoyov fityi&tj ■ 
Xoyov xaXeia&a. 



ro def. 120, 1. Barluam logiBt. I def. 1. 

Barlaaoi I def. S. 3. Hero dnf. 12T, Paelli^ 
ro dcf. 123, 1. 5. Hero def, 121. 



Hero def. 121. 
p. 8. i. H( 
def. 124. 



1. SpotJ oia. PBFp. nmiiflrOB om. codd. omnea. 
Tov Hero. 4. iXuaeovos V, ut lin. 6. 7. xolu\ P, Hero^ 

n^oe nJtlqla noui Theou^(BFV p),^ CampaDUB. Poat axioi 
add. avaloYia di n rmv Hyaiv tavtotijt Bp, Campanue; mg. 
in. 2 F Y; mg. bU m. 1 et m. 2 F; om. Hero. 8. /jew] 



Liber V. 

Definitiones. 

Pars est minor magnitudo maioris, si maiorein 
r. 

Multiplex autem maior est minoris, si minor 
letitur. 

Ratio est duaram eiusdem generis magnitudinum 
um quantitatem quaelibet habitudo. 
Rationem inter se habere magnitudines dicun- 
ae multiplicatae altera alteram superare possunt.. 
In eadem ratione magnitudines esse dicuntui 
ad secundam et tertia ad quartam, si primae 
iiae aeque multiplices secundae et quartae aeque 
»lices aut simul superant aut simul aequales ' 
lut simul minores sunt suo ordine^) sumptae. 
Magnitudines autem eandem rationem habentes 
rtionales uocentur. 



Hoc est: ita ut coniuDgantur prima secundae, tertia quar- 
respondeat loco et ordine prima tertiae^ secunda quartae. 

si Ma ==,Nb et simul Mc ^ Ndy erit a : h «^ c : d. 

mkel: Zur Gesch. der Mathemat. p. 390. 

a m. 1 P. 9. vnsQsxsi^v] -siv in ras. V. 14. noXka- 
lafimv P, corr. m. 1. 15. vnsQixsi B. 17] supra m. 
iXXetnsL B. Xritpd^ivxa] -ri- e corr. m. 2 V. Deff. 
«rmutauitP; ut nos BFVp, Campanus; ex Herone nihil 
di potest. cum etiam def. 8—9 ante def. 7 habeat. 
l;UoyTa Xoyov fisyid^] loyov ?;uovTOf (isyi^ F; fx^^^^ 
7 Xoyov y. avdXoyov'] Xoyov avaXoyov post ras. 7 litt. 
. transit m. 2 F. 



1 HTOlXEiaN e'. I 

"Eczca onoOttovv (iiyi&tj tu AB^ r^d onoStovovv 

jisyid^mv Ttov E. Z taav ro Jil^&^os ixastov exaSrov 

iaaxig itokXaTtXK&tov ' Xtya , oti oOanXiioiov ioti zo 

^■IB Tov E, ToOavTanXdoia ietai. xtd ik AB, TJi 

5 Tav Ey Z. 

'E%t\ yc/Q ioaxis ^UtI noXXanXaOiov zb AB Tov 
E xal ro Fd tov Z, aoa apa iOTlv Iv rp AB fie- 
ys&Tj lOtt Tw E, toeavza xal iv tip V^ iVa tkp Z. 
Si,riQriO&fa zo ^ev AB els ^a rp E (leyd&tj tea tu 

10 AH, HB, t6 tfi r^ flg ra zip Z iott to. r&, ®^- 
lezBi dij leov t6 itXij&os' zmv AH, HB tS mXjj&st 
rmv re, &J. xal inel fffov iazl z6 filv AH 
E, zb Sl r& T{5 Z, loov apa ro AH Ta E, xal 
AH, r@ ToEg E, Z. dta tk avta dij i'0ov eOTi 

15 HB tp E, xttl ra HB, @J rofg E, Z' oOa Spa iezh 
iv TM AB l'6a Tm E, Toeavta xal iv Tofg AB, T^ 
laa tots E, Z' baanXaeiov opa iorl rb AB tou E, 
ToattVTanXttOi.a lOtai xal ta AB,r^ tav E, Z. 

■ 'Eav aQtt ri baoeaovv (isyi&tj bnoeavovv fieyi 

20 &Sv toeav to jrA^frog fxaffiov exderov lottxtq noXXt 
aldoiov, oOanXdatov ieriv h/ tiSv fieyt&mv ivog, 
aavraitXixaia iarai xal ta advra rmv JtdvrstV Sxt 
e$£i Seilai, 



'Eav 



• devt, 



lolaxldinov P. 



y itoXXaxXti- 
7. lctiv] fieyf&i) 



i 



E, 

i 



9. xm] corr. i 
fittx] corr. esoieu m. 1 V. 10. tls] tt p. t^] eorr. 

ex z&v m. l B. 11. (eov\ m. 2 V. AH, MS] Pip; re, 

e^BVp. 12. FG, eJ] V<p;AH, HB BVp. foo*] m. 

2 V. 14. id] in ras. p. Emendatio ed. Basil. lin. 13i tau 

Sfu -xctl za AH^ re lois E, Z et lin. IC: xal lo 6 J im Z, 



ELEMENTORUM LIBER V. 9 

li quotlibet magnitudines AB^ F^ quotlibet 

magnitudinum E^ Z nu- 

? B ^' ^ mero aequalium singu- 

lae singularum aeque 

maltiplices. dico, quoties 

lex sit AE magnitudinis E. toties multiplicem 

B + r^ magnitudinis E -^- Z. 

n quoniam AB magnitudinis E et FA magni- 

Z aeque multiplices sunt, quot sunt m AB 

iTidines magnitudini E aequales, totidem etiam 

' sunt magnitudini Z aequales. diuidatur AB 

poitudines magnitudini E aequales AH^ HB et FA 

.gnitudines magnitudini Z aequales F&^ @A, 

numerus magnitudinum AH^ HB numero 

tudinum FG^ GA aequalis erit. et quoniam 

= £ et r® = Z, erit AH = £ et AH + F© 

+ Z. eadem de causa HB = E et HB + GA 

]- Z. itaque quot sunt in AB magnitudines ma- 

ini E aequales, totidem etiam sunt in AB -{■■ FA 

tudiui £ + Z aequales. itaque quoties multi- 

;st AB magnitudinis E^ toties multiplex erit 

AB + rj magnitudinis £ + Z. 

go si datae sunt quotlibet magnitudines quotlibet 

tudinum numero aequalium singulae singularum 

muItipliceS; quoties multiplex est una magnitudo 

, toties etiam omues omnium erunt multiplices; 

erat demonstrandum. 

II. 
prima secundae et tertia quartae aeque multi- 



a xal Ttt HB, SJ necessaria non est. 21. iart V. 26. 
ov qp (non F). 



10 



ETOIXEiaN I 




Stov xal tqCtov rsraQTOv, tj di xal JiifiXTOV 
SsvT^QOv iuaxis ^oi.X«jcXdotov xul txTov te~ 
rapTov, xal avvttQiv tcqiStov xal xiftitTov 
dsvTSQOV ieaxig SiSTat 7ioli.anXaeiov xal TQt- 

5 TOV Xal tXTOV TtTKQTOV. 

/Zpmiov yaQ to AB devT^QOV roi; F iaaxts Itn 
TtoiXajtXaewv xal tqCtov zo dE zbtkqtov rotJ 
isra di xaX nefintov t6 BH dtuTtQov tou F iaaia 
^oXkajtXdeiov xal ixrov ro E& tixdQTOV tov Z" Xiya 

10 oTi xai avvre&Ev itQtatov xal aijiJiTov ro AH devi 
Qov zav r iedxig lOzat TcoXlaitlaaiov xal XQitov i 
ixxov To /i® zezdffzov zov Z. 

'Exsl yccQ iadxiq iexl aoXi.an.Xdtttov x6 AB \ 
r xttl x6 AE xov Z, oea apa iaxlv iv rra AB fal 

15 Tt5 r, toffaiiia xal iv x^ ^fE tea rp Z, Sik < 
ttvTtt di} xttl oea iotlv iv Ta BH iea ta F, i 
xal iv za E& lOa t^ Z- oOa aQa iotlv iv oX^ i 
AH lea rp r, zoaavxa xal iv okip za ^® laa ip Z' 
oOttTtldaiov &Qtt iatl To AH zov F, zoaavTKXkdaiov 

20 iazttL xttl ro /]& zov Z. xal awz8&lv «pc nQazov 

xal Ttinnzov to AH Sevzbqov tov F iadxis laxtti 

itollaKXdoiov xttl rptrov xal Exrov zb A& zBzdQzov zov Z. 

Eav ttQtt itQatov dsvziQOv iOttXig tj noXXttnXaaiov 

xal tqCxov zbtkqxov, {j Si xal Tti^Ttzov devTiQov 

35 iadxig jioXXaTtXdoiov xal iXTOv TETdqxov, xaX awts&^v 
TiQmxQV xal %i(LnTov SsvztQov iOttXig iazat noXXaaiA 
Oiov xttl xQtzov xttl txvov terdQtoV onsQ ISst Sst^ 



6. ievTiffOr 



corr. ei SevTtqov Y. 1 
! P. 17. E©] EBip. 
Z] corr. ei r m. 1 P. 
21. ieTUi] fffcco B, iffci 



. ioTiy P. 
18. je] corr. 



ELEMENTORIIM LEBER V. H 

plices sunt^ et quinta secundae sextaque quartae 
aeque multiplices^ etiam prima quintaque compositae 
secundae et tertia sextaque compositae quartae aeque 
multiplices erunt. 

nam prima AB secundae ^et tertia ^E quartae 

Z aeque multiplices sint^ et ^inta BH secundae F 

3 jf sextaque E& quartae Z 

"^' ' ' ' ' ' aeque multiplices sint. 

Ti — I dico, etiam primam quin- 

jg Q tamque compositas AHae- 

' ' ' ' ' ' cundae F et tertiam sex- 
^' ' tamque compositas ^& 

quartae Z aeque multiplices esse. 

nam quoniam AB magnitudinis F et ^E magni- 
tudinis Z aeque multiplices sunt, quot sunt in AB 
magnitudini F aequales^ tot etiam in ^E sunt magni- 
tudini Z aequales. eadem de causa etiam, quot sunt 
in tota BH magnitudini F aequales^ tot etiam in 
E& sunt magnitudini Z aequales. quare quot sunt 
in tota jiH magnitudini F aequales, totidem etiam 
in tota ^& sunt magnitudini Z aequales. itaque 
quoties multiplex est ^ff magnitudinis F, toties multi- 
plex erit etiam ^& magnitudinis Z. itaque etiam 
prima et quinta compositae AH secundae F aeque 
multiplices erunt ac tertia sextaque 2J& quartae Z. 

Ergo si prima secundae et tertia quartae aeque 
multiplices sunt, et quinta secundae sextaque quartae 
aeque multiplices, etiam prima quintaque compositae 
secundae et tertia sextaque compositae quartae aeque 
multiplices erunt; quod erat demonstrandum. 




'Eav srpratov StVTiQOV ladxig ^ noMajlkA 
0I.OV niil Tffirov TszaQTOv, Xi}tp&fj Sh lodxt\ 
nokkii3ti.ix6ia TOt) T£ 3r()(DTou xai rQiTov, xkX 
3 Sl ieov rav Xt}(p^vriov ixdrEQOv Bxaripov 
iedxtg larai noXXiniKdaiov t6 jiiv rou dEvri- 
Qov ro 6i rov rfrR^roti. ^H 

IlQcorov yiiQ ro -^ dtVTEQOV roti B {adxis fiJTia^H 
noXXanXdaiov xal rptroi' ro f TirHprow row ,i/, xol'^^ 
10 e^A^qi^dra rtov j^, J' iadxis moXXa^Xiieia za MZ, H®' 
Xdya, ort iadxts iarl noXXa^^Xdeiov ro £Z roiJ B «al 
ro H0 rov -^- 

'Sjtft yap Cadxis iOtX noXXanXdaiov ro £Z ro« 

A xal rh H@ tov V. Saa aQa iarlv iv t^ EZ Ajo 

15 tM A, toaavra xal iv ra H& taa ta F. diriQrja&ta 

ro iilv EZ eig ta ra A fiiyi&T] taa rd EK, KZ, rc 

di H@ sis ra rc5 F iaa ta HA, A@- gatat Si} taov 

ro nXij&og riav EK,KZ rro nXti&Ei tcov HA,A@. 

xal ind Cadxig iarl noXXanXdeiov ro A rov B xal 

ro r toi A, i'aov Sh ro [isv EK ta A, rh Sh HA 

ra r, Cadxig apa iarl :toXXanXdai.ov ro EK row B 

xal rh HA roii A. Sicc r« avra dij iadxig iarl 

jioXXttnXdaiov rh KZ tov B xal ro ^® tov jd. i«^^ 

ovv Jipciroi' ro EK SEvtipov rot' B iadxtg iatl KoAJlv^ 

t 26 «Xaatov xal tptVoi' ro HA rfraprow rou ^, lan i 

xel nd[inrov rh KZ StvtiQov roij B ladxig noill»' 

nXdaiov xal £xrov ro A& rfitfproi/ rov .d, xttl ifuv- 



iarl 
:«■ 



4. Tf] om. BVp. 10. slXijqiS-aieav p. 11. laanis l«tl 
jioXXaKiaatov\ oaani.aaiov P. B] in ras F. 14. in{n\ 

gnpra F. taa] m. 2 P. 15. -Aal] S>i v,al V. 16. lo] m. 2 V. 

E^e lo] in rae. m. 2 V. 20. J^] (prius} aiiprft m. 1 comp, V. 

22. ^otC* P. 23. Toii J] poBtea add. F. 25. hxiv 



^ 



ELEMENTORUM LIBER V. 13 

m. 

Si prima secundae et tertia quartae aeque multipli- 
ces sunt; et primae tertiaeque aeque multiplices sumun- 
tur^ etiam ex aequo^) magnitudinum sumptarum altera 
secundae altera quartae aeque multiplices erunt sin- 
gulae singularum. 

A\ 1 Nam prima A secundae £ 

2, et tertia F quartae ^ aeque 

K zsint multiplices, et sumantur 

^' ' « ' magnitudinum A^ F aeque 

r ! ^i multiplices EZ^ H0. dico, 

^, I EZ magnitudinis B et H0 

A magnitudinis z/ aeque multi- 

^' ' '® plices esse. 

nam quoniam EZ magnitudinis A et H® magni- 
tudinis T aeque multiplices sunt, quot sunt m EZ 
magnitudines magnitudini A aequales, totidem etiam 
in H% sunt magnitudini F aequales. diuidatur EZ 
in magnitudines magnitudini A aequales EK^KZ^ et 
ff® in magnitudines magnitudini T aequales HA^ 
AS. erit igitur numerus magnitudinum EK^ KZ 
numero magnitudinum HA^ AS aequalis. et quoniam 
A magnitudinis 5 et F magnitudinis A aeque multi- 
plices sunt, et EK = A^ HA = F, erunt EK magni- 
tudinis B et HA magnitudinis A aeque multiplices. 
eadem de causa KZ magnitudinis B et A0 magni- 
tudinis A aeque multiplices sunt. iam quoniam pri- 
ma EK secundae B et tertia HA quartae A aeque 



1) Hic non proprie ad definitionem rationis di* tcov (17) 
respicitur. 



«al fxzov ^i^l 



14 STOIXEIflN *', 

Tc9iv apc nQmrov xal Tiifintov ro EZ Sevzipov i 
B ioiixis ^Stl noXla7ti.detov xal rpiTov xal i 
H® TStaQTov Tou jd. 

'Eav uifu nQmxov SimiQOV iedxis t] nolXaaXdotmf 

5 xal tqCtov reraprov, lij<pd-f] di roi5 ngmzov xal rptVou 

ladxtg noXXtticldeiu , xal dt' i'<fov xmv Xi}ff>9bvzaiy 

ixdztQov ixaripov (isdxis Iotki nolKankdotov to ^%% 

rov dsvriQOv ro dl tou r£TapTow oneQ iSat Sel^m 



10 Eav nQarov npbg dEVTSffov tov «i 

loyov xttl rpizov nQOg ritaQrov, xal r 

nokXanXdaia rot' te aipmxou Xttl Tptrow ii^Ofl 

za ladxis noX?.anldeia tov SsvriQov 

Tdgzov xa&' onoiovovv noXXanlaOiaafiov tbm 

B KWToi' f^£i Xoyov Xrj^&ivTa xardlXtjXa. 

TIqcotov ytt(f To A aQog StvrtQOV t6 B i 

Tov txiTM loyov xal tqltov io F TiQoq TBTaqtov ro 

^, xttl eif.riip&<n rmv (ilv A,r iadxig nolXaxXdata 

To E, Z, trof 6i B, id aXXa, a ^zv%bv, iedxis «oAAo- 

20 aXdaia ra H, &' liyco, oti iarlv rog ro E nQog i 

H, ovtme ro Z jrpog ro @. 

HAij9)*ra y«p rroi' fih» B, Z /ffaxig moAAiaCiUeilMl 
r« J^, ^, tav Si H, & aXXa, a Jruj^ev, ledxig noXXm 
nldeia ra M, N. 
& [Kal] insl iedxtg iotl noXXanXdeiov ro ft^ 

6, 3i leiitne Ttollaxliiaia tov ttgatov xal Tfit 
p. 12, 3-4. 8. ieiiai] noivaat V. 18. TJ corr, es 
*«] pOBtea add. m. 2 F. a) m. 2 F. 20. iaiiv] om. V. 21 
H htiv ¥. 23. allff, a ftujjr] mg. m. 2 V. S] aapra F 
24. W"] in ra». m. 1 p. 26. vai] m. 2 P. 



ELEMENTORUM LEBER V. .15 

multiplices sunt^ et quinta KZ secundae B sextoque 
jiS quartae ^ aeque multiplices sunt^ etiam prima 
quintaque compositae EZ secundae B et tertia sex- 
taque compositae HS quartae ^ aeque multiplices 
erunt [prop. 11]. 

Ergo si prima secundae et tertia quartae aeque 
multiplices sunt, et primae tertiaeque aeque multiplices 
sumuntur^ etiam ex aequo magnitudinum sumptarum 
altera secundae altera quartae aeque multiplices erunt 
singulae singularum; quod erat demonstrandum. 

IV. 

8i prima ad secundam eandem rationem habet 
ac tertia ad quartam, etiam primae tertiaeque aeque 
multiplices ad^secundae quartaeque aeque multiplices 
qualibet multiplicatione productas eandem rationem 
habebunt suo ordine sumptae. 

^i 1 Sit enim A:B = 

B i JT : ^, et sumantur 

E , i magnitudinum A, F 

H\— I — — 1 aeque multiplices E, 

K 1 Z et magnitudinum 

M\ ! 1 By /1 aliae quaeuis 

r. : aeque multiplices Hy 

ji — i ®. dico, esse E : H 

Zi i 1 = Z : 



Si — i — I — ; sumantur enim 

ji\ 1 ! magnitudinum £, Z 

jVTi , 1 1 aeque multiplices Ky 

A et magnitudinum /f, ® aliae quaeuis aeqae mul- 
tiplices My N. iam quoniam E magnitudinis A^ et Z 



STOIXEmN e'. ^H 

' A, ro 6b 2 Tow T, wl tlltiaTdb zav B, Z iad- 
xig nollaJtkatSicc ta K, A , iadxig apo: iatl noXXanXu- 
eiov To K xov A xal ro A rov r. Sia Ta avta 
di] isdxig eazl jtoi.Xaxkaei,ov ro M row B xal to W 
5 Tov A. y.ai ^tcbC iauv ag to A npog lo B, ovTmg 
TLi r jTpog ro ^, Xttt (Wt;WT«i zmv fi£V v^, T ^oaxtg 
itoiXaitXdeitt ta K, A, rav Se B, A aXXa, « hv%sv, 
ieaxig TioXXanXttOia za M, N, sl ffpa V7lBQi%ei. ro K 
tov M, vjtepdxtt xal ro A tov N, xal si Ceov, ioov, 

10 xal bI IXartov, tXarrov. xai ioti ra fiiv K, A xmv 
E, Z iedxis aoXkaxXdei.a, r« dl M, N tmv H, ® aXXa, 
a hvxBV, ladxig jioXXairXdaia' Seriv a^a atg ro E w 
%Qog To H, ovztog ro 2 jrpog ro 0. ■ 

'Eav UQa TtQmrov jrpog dsvTB^ov tiiv avrov ^XfA 

15 Xoyov xal zqitov irpog rErccprov, xal ro; iedxtg TtoXXu- 
jtXdaitt roti T6 mpoirow xttl rgitov jrpog r« iedxig 
ntoXXaaXdaia tov Ssvriqov xal reraprov rov curoi' 
e%BL Adyof !(«#'. bnoiovovv xoXXaitXaeiaefiov Aij^OeWib J 
xardXXijXa- ojtBQ sSbi. dBt^ai. 



20 



'Eav jidyE&og HEyd&ovg iedxig ^ ■icoXX 
JtXdttiov, ojtBif d(paiQB&%v d^aiQB&ivzog, 
ro Xoi,xov roiJ Xomov iedxig lerai jtoXXa 
Gbov, oeaaXaetov iari ro oAor roti oXov. 



]rr. m. rec. 2. jioUaJilaoto*] ; 

1. 5. oSto> F. 6. fiei'] on 

10. PoBt fka-zTov in P repetimtm 
loi M xoi TO A Tov N «b! e( taov t 
loTiv P. _ A}e corr. m. 2 : 
«] auiira m. 2 F. 18. ti afiotov] tfTtt)}vov y (aonl 

17. xo:&' onotovotii' noXIanZnoiuafiDi' tov ecvxhv i^ci Xoyov F 
cfr. p, lilin. 14— 15. 19. ilftgttil corr. exjioi^oat V. Deinde ^ 
Theoni ixei ovv ISeCx^fj, oti, sl vmeix^i lo K tov M, 



7. «] Bupra F, 
^jid ^jiseE);;! tD 



ELEMENTORUM LIBEK V. 17 

magmtadiiiis F aeqac multiplices sunt^ et sumptae 
jrant magnitadimmi £^ Z aeque multipiices Jl, yi« eiit 
K magiiitadiiiis ^ et w^ magnitudinis F aeque multiplex 
[prop. iilj. eadem de causa M magnitudinis B et AT 
magmtadiniB ^ aeque multiplex est. et quoniam est 
AiB = ri^j et sumptae sunt magnitudinum A^ F 
aeqae multipUces JT, A et magnitudinum By ^ aliae 
qoaeais aeque multiplices M, iNT, si iC magnitudinem 
3f superat, etiam A magnitudinem N superat^ et si 
aequalia, aequalis est, et si minor, minor [def. 5]. et 
Kj A magnitudinum E^ Z aeque multiplices sunt^ My N 
autem magnitudinum Hy S aliae quaeuis aeque multi* 
pliees. itaque E: H = Z:G [def. 5]. 

Ergo si prima ad secundam eandem rationem 
habet ac tertia ad quartam, etiam primae tertiaeque 
aeque multiplices ad secundae quartaeque aeque multi- 
plices qualibet multiplicatione productas eandem ratio- 
nem habebunt suo ordine sumptae; quod erat demon* 
strandum. 

y. 

Si magnitudo magnitudinis aeque multiplex est 
atque ablata ablatae, etiam reliqua reliquae aeque 
multiplex erit ac tota totius. 

xal t6 A xov N, %ai sl Caov Ccov^ xal si iXatxov ilaxxov^ 
dfjjlov OTt xal €^ vxBQixsi xo M xov Ky vnsQSxsi, xal t6 N xov 
A, xal st taov taov^ xal sl iXazxov ^Xarrov, xal dia tovto 
^tfTttt xal mq xo H nQog xo E, ovxag x6 S UQog xo Z. IIoQtaiia, 
i% d'^ xovxov (pavsQov , OTi iav xiaaaQa fisyid'!! dvdXoyov y, 
%al dvdnaXiv dvdXoyov iaxat (FBY; primum oTi om. B; 
ovxm pro ovxmg F; semper iXaaaov V; in p non ezstant nisi 
ultima inde a noQiaaa); idem in P. mg. m. rec. (om. priore 
OTi); om. P m. 1, Campanus; cfr. ad prop. VII. 24. to] 

corr. ex tov m. 1 F. t6 oXov] supra p. 

Euclides, edd. Heiberg et Menge. U. 2 



18 STOIXElJiN i 



fuQ t6 AB jifyB&ovg tou F^ Cffttxiq 
itfrro TtokkttnkaiStov , onep KipKiQB&lv z6 AE ^ipaigS' 
^iVros Tou fZ* Xsya, ort xal Aoiffox' t6 £^5 loiTiov 
tov ZjJ Cauxis eStai iioi.Xanka0tov, oOaiti.aOi6v iffziv 
oAoi' t6 AB o^ov tow r^. 

'OffaaXdoiov ydff iiSTi t6 AE tou TZ, roOwi;- 
rajr^Kfiiov yEyoviz(a xal t6 £B toi' f/f. 

ATtti iitsl Caaxig ifftl itolXanXuUiov t6 AE toii 
rZ xai t6 EB ToiJ i/r", Cedxig «pc /flTi :iTo/AanAa- 

10 aioK t6 AE tov rZ xal ro AB zov HZ. XBlzai 8h 
iadxig jtoHanittSiov zb AE tov FZ xttl t6 AB tow 
F^. Cadxig Sga iarl TtoXXaaXdaiov zo A B ixariQov 
rav HZ, Fzt- fffov aga to HZ rra F^i. xotvbv ayjj- 
pijc^© 10 rZ' lombv aga zb HF Xomm r^ ZjJ 

15 i!aov iativ. xal iael Cadxtg ietl noXXanXdeiov %b 
AE xov rZ xttl 10 EB zov HF, taov Si zb HF tc5 
jJZ, iadxis apa iatl naXXaxXdaiov t6 AE rov fZ 
Hal tb EB zov Zz/. iadxig 61 vaoxtitai xoXXanka- 
awv tb AE Tow rz xal t6 AB zov VA' Cadxig Sgtt 
iatl TtoXXamXdaiov ro BB tov Z^ xai, tb AB tov 
PA, xal Xoixhv ttQa t6 EB XotTtov zov Z^ iadxi 
iatat noXXanXdaiov , baanXdaiov iaxiv oAov t6 A. 

o^ou TOTJ r^. 

'Eav aQtt [liys&os ittyi&ovs iedxts tj aoXXaaXaaiovy 

£5 oasQ dipai.(fE&hv ttfpttiQE&ivzog, xal rb Xotnbv tou 

Aoiiroi? COttxts iazat itoXXajrXttaiov^ baaxXdatov iffUr. 

xal tb o^ov Tov oAov oxbq idst Set^ai. 



xis^9 



ov 



i. ZJ] JZ Bp; Z^, seq. rtta. 1 litt. et Z in raa. V; E 
in nu. F, imv} hti to F. 6. Ioti] iaxiv alov, deiel 

oXov V. 8. Kol iictl— 9 : HVl om. p; mg. m. 2 B. 
EB] B in rfti. P. ifr] corr. m, 1 ei TH V; TH B; riT 




ELEMENTORUM LIBER V. 



19 



Sit enim maguitudo jiB magDitudinis P^ aeque 
E multiplex atque ablata jIE 

~' ablatae FZ. dico, etiam reli- 
-I — i-^ quam EB reliquae Zz/ aeque 

muitiplicem esse ac totam ^B totius F^. 

uam quoties multjpleif est j4E m^nitudiniB FZ, 
totiea multiplex fiat EB magnitudinis FH. et quo- 
niam ^E maguitudinis FZ et EB magnitudinis HF 
aeque multiples est, etiam AE maguitudinis FZ et AB 
magnitudinis HZ aeque multiples erit [prop. I]. et 
posuimus AE magnitudinis FZ et AB magnitadiuis 
r.J aeque multipUces. itaque ^B utriusque HZ, Fjd 
aeqiie multiplex est. quare HZ = FjJ. subtrahatur, 
quae communis est, FZ. itaque HV =^ Zz/. et quo- 
niam AE magnitudinia FZ et EB magnitudinis HF 
aeque multiplex est, et HF=^Z, erit AE m^ni- 
tudinis rz et EB magnitudinis Z^ aeque multiplex. 
Bupposuimus autem, esse AE magnitudinis FZ etAB 
magnitudinis FJ aeque multiplicem. itaque EB magni- 
tudinis Z^ et AB magnitadinis F^ aeque multiplex 
itaque etiam reliqua EB reliquae Z -J aeque mul- 
tiplex est ac tota AB totiua FJ. 

Ergo si magnitudo magnitudinis aeque multiples 
est atque ablata ablatae, etiam reliqua reliquae aeque 
Diultiples erit aa t«ta totius ; quod erat demon- 
strandum. 



1 



?. 10. AB] B va ras. F. HZl in raa. BPV. 12. 

' iniv P P. 14. Zj}] P, Pm. li JZ BTp, Fm. 2. Ifi. 

' iniv) P; comp, p; iffi/ BPV. «oHcuclaaisiv ip. 16. HF] 

I (priuB) seq. raa. 1 litt., H ic rHB- V. 17. JZ] Z^ P. 20. 
' lariv P. ZJ] 9, .dZ F. 86. ^OTi* P. 



ETOIXEIJiN E 



'Eciv 6vo (isyd&7} 
]toli.a^i.deia, xal uipai 
Ceaxig rj aof.i.ajtJLiiata, 
j ijrot fffa iariv 7) lad: 



ivo ^eyt&mv faaxig 
ft&dvra rivtt Ttov ai^Tioi^ 
«ttl Tu loLita zotg avrots 
■.ig avtt^v itollajti.ttata. 




^vo ydp [i.tye9ti ra AB. F.^ rfvo fuye&av t 
E, 2 laaxis lara jtoAAajrAweto:, xal a<fittiQt%dvra 
j4H, r&rmv avtiav rav E, Z iadxig ieta iitolXajtlar'^ 
aia- Isya, ori xal lonta ra HB, ®J TOiq E, Z ^i 

3 tea iatlv 7/ tadxis avtmv aokkankdaitt. 

"Eozta Y«Q agoztQov to HB ra E laov i.iyt^[ 
ort xttX tb ®^ ra Z iaov iativ. 

Ktia&a y«Q ra Z taov ta VK. intl 
iati itoKXaitkdfSiov th AH tov E jtaJ v6 F® rov 

s [aov 8i 10 fiiv HB ta E, th S\ KF ra Z, ledxts 
iepa iatl aoi.laitldeiov tb AB tov E nal ro K@ 
Toij Z. iadxig ds vnoxtirai nokkankdatov 10 AB 
tav E xal 10 FA roi' Z' ladxig apa iarl Jtokka*' 
ntldaiov ro K& roii Z xal t6 Fi^ TotJ 

ovv EXttttQov rmv K&, VA row Z ladxis iutl JtoJika- 
nkdaiov, raov «Qa iazi rb K& ra FA. xoivhv 
dtfr^Qria&a zb r&' AoiTthv apor ro KF Xotna Tra &^ 
taov iativ, dkktt tb Z ta KF iariv teoV xal to 
®^ apa ra Z tUov iariv. mate ti ro HB ta 

5 fffov iariv, xai tb @A Caov larat ra Z. 

'Oftoimg Sij Sti^ofitv, ozi, xav nokkankdaiov §' 



■MtS 

K9 
I ^B 

vov -^ 
S.d 



7. IffT 



12. @j] jep. z^ z] Q 

BV. 13. ral corr. es id 1 
2 P. xal WV. 14. loE 



r 



ELEMENTORDM LIBER V, gl 

VI. 

i diiae magnitiidiiies duarum magiiitudiimm aeque 
multiplices sunt, et ablatae quaeuia magnitudinea 
earundem aeque multiplices sunt, etiam reliqnae iiadem 
aut aequales aunt aut aeque earum multiplices. 

Nam duae magnitudinea ^B, Ti^ duarum magni- 
tudinum E, Z aeque sint multiplices, et ablatae magni- 
tudines AH, F® earundem E, 2 aeque muitiplices 
aint. dico, reliquas HB,&^ aut aequalea esae E,Z 
aut aeque earum multiplices. 

nam prius sit HB = E. dico, eaae etiam &^ = Z. 
ponatnr enim FK = Z. quoniam jiH 
magnitudinis E ei r& magnitudinis Z 
aeque multiplex est, et HB = E, 
XK-Vk-h-T-i^ Kr=Z, erit -4B magnitudinis £ et 

Zi-i x& magnitudinis Z aeque multiplex 

[prop. II]. et suppoauimus, esse AB magnitudinis 
Eetr^ magnitudinis Z aeque multiplicem. itaque 
K& magnitudinig Z et TW magnitudinia Z aeque 
multiplex est. iam 'quoniam utraque magnitudo K&, 
FjJ magnitudinia Z aeque multiplex eat, erit K® = Tz/. 
Bubtrahatur, quae eommunia est, r&. itaque Kr= &^. 
sedZ = Kr. quareetiam®z* = Z. itaque si /f-B = E, 
orit etiam ®J = Z. 

similiter demonBtrabimus, si HB magnitudinia E 



A^ ,_ 

Ei— I 



ie^iv P. 18. rol toS V, 


con-, m. 1; om. q> (non F). 


iniv P, 23. rol P jn. I , F 


m, 1, Bp; Tol Pm. 2, F m. 2, 


V in rafi. m. 2. Z] KF V. 


™] Pm. 1, Fm. 1, Bp; to 
'^r] Z V. rd]TmBp. 


Pm, 2, Fm. 2, V iii rae. m. 2, 


24. ej] i/e P. ira] 10 


Bp. ieov lBziv\ PB; hxtv 


fto^ FVp. tl\ P; S« The 


on (B,p Vp). 25. iaxCV\ -,* 


in me. P; ^oti BV; comp. p. 


nttl 10 ej leav Ux<xi\ mg. P. 



ETODCEIiiN t 



TO HB tov E, roaamaalaeiov iCrui xal rh @ 
roiJ Z. 

'Eav aptt dvo (leyE&rj dvo (isya&av {<fdxis y «oKK»-- 
xXdsia, xal a<paip89tvztc riva rmv ctvTiav iaaMg y 
5 aoXlanldaiR, xal tk koink toEs avrol^i ^rot fffa itftlv 
^ iodxig avTtov nolltt^iidoitf oncg eSit dst^i. 

&'. 
Ta laa XQog to ecvtb toj/ avtbv ijf 
yov xal 10 auTo jrpog zd iSa. 
ti "Eaza Tffa (lEyd&ri td A, B, «AAo ii ti., b ETV^fvJ 
(isys&og to r"" AEya), ort ixdtEgov t£v A, B Kpog zh^ 
r TOi- auTOi' ^x^t loyov, xai ro f Jtpog exattffOt 
tmv A, B. 

Eil^qi&a ydg tav (ilv A,R iadxig 7ioli.aicldatii 

5 ra A, E, tou di F allo, o Itvxsv, noXXaxldeiov % 

'Ejctl ovv leaxig iatl nokkanldatov tb A xov . 

xal To B Tou B, teov dh ro A tS B, laov «pa Kal 

ro j^ ta E. allo di, o hvxfv, to Z. Ei apa 

vjttQixtt To A Tov Z, vjttpix^t, xal tb E tou Z, xal 

20 ei fffov, i'ffoi', Ktt! si IXaTtov, ^attov. xai istt tk 

(»iv /i, Etav A, B iadxig noXXaaXdeta, ro Si Z tov 

r aXXo, o Itvxev, noXXanXdatov iattv apa d>g i 

jrpos To r, ovTiaq ro B Jtpog to F. 

Aiya [d^j, ori xal to E nQog ixdtCQOV Ttov A, J 
S6 Tov avtov extt Xoyov. 



5. nal la] la in las. P. iazh] corr. ei forui p, 

3] eapra m. 2 F. fiu^f Yp. 1*. fi^f] PF; om 

16. 31 Bupra m. 2 F. frfx* Vp. IG. to] to5 i 

Tov] corr, ei ro P. 13. o] m. 2 F. fruif Vp, 

raa, 1 litt. F. 20, KBt'] comp. F, deiii add. xai q>. 



1 



ELEMENTORUM LIBER V. 23 

moltiplex sit, aeque multiplicem esse O^ magni- 
tudinis Z. 

Ergo d duae magmtudines duarum magmtudinum 
aeque multiplices sunt, et ablatae quaeuis magnitu- 
dines earundem aeque multiplices sunt, etiam reliquae 
iisdem aut aequales sunt aut aeque earum multi- 
plices; quod erat demonstrandum. 

VII. 

Aequalia ad idem eandem habeAt rationem et 
idem ad aequalia. 

Sint aequales magnitudines A^ B et alia quae- 

A\' — H jd\ \ 1 1 1 uis magnitudo r. dico, 

Bi 1 Ei 1 1 , 1 utramque magnitudi- 

Ti 1 z I 1 1 1 nem A^B 2lA F eandem 

rationem habere, et F ad utramque A^ B. 

sumantur enim magnitudinum A^ B aeque multi- 
plices A^ E^ et magnitudinis F alia quaeuis multi- 
plex Z. iam quoniam A magnitudinis A eti E magni- 
tudinis B aeque multiplex est^ et A = B^ erit etiam 
A = E. et alia quaeuis magnitudo est Z. itaque si 
A magnitudinem Z superat, etiam J^ magnitudinem Z 
superat; et si aequalis, aequalis est^ et si minor 
minor. et magnitudinum A^ B aeque multiplices sunt 
A, Ey et Z magnitudinis F alia quaeuis est multi- 
plex. erit igitur 

A:r=B:r[dei.5]. 

dicO; etiam E ad utramque magnitudinem A, B 
eandem rationem habere. 

corr. p. 21. Z] EZ F. 22. o] om. F; add. m. 2 euan. 
irvxs Vp. iaziv] bis P. 24. di}] om. P. 



24 



ETOrXEISN E 




Tav yag aVTwv xaTaaxevaeQivtav bfioiiDS i 

lofifi', OTi i'6ov /ffr! EO z/ Tta £■ kZ^o ds' rt ro Z* 

i/ npa vXfQixti to Z row ^, imfQdxfi ««I rou £, Jtorl 

ft Ceov, ieov, xal fl IXaTTOv, sXctTrov. xaC iUTt ro 

' filv Z ToiJ r nolKaTtXaatov , t« Sl z/, £ tiow ^, B 

r JTpog To ^, ovrag ro T jrpog ro B. 

Ta laa a^a jrpog to avTO rof avTov tx^t i.6yon 
xttl To «wro Kpog t« i'ffa. 

J n6gi6(ia. 

'Ex Sij Tovtov (pavBQ^v, azi iav (tfyi&ri t( 
Aoyof fj, xni avttnahv avaXoyov EffTRt. ojrt p fdf 






Tav avi6mv (i.eyfd^av ro fifti;'"' ^P° 
'j BWTO fif^Jova AoVof fxsi ^:XfQ to /AnTTOVjl 

jtal ro avTo arpog ro cXarrov fici^ova la 

JXft ■^Mtp iTpOg TO ftfEgoV. 

"Eerfa aviaa [ifyiQ'j} rit AB, F, xal faro) (ift^oi 

To j4B, aAAo tfe, o Srvj^fv, ro iJ' liya, ot( t6 .^] 

20 JEpos ro ^ fLfitfiva Xoyov ixfi- TJacp t6 r* Jtpos ro 

xal To ^ Jrpog To V fiiC^ova Xoyov 't%fi- ^aiEp npj 

t6 v^B. 

'fiwei yap ft£f£oV iffrt t6 .rfS roiJ J", xfia&at 
r iaov zb BE- rb S^ ilaeaov rrov AE, EB «oL 

nl p. 1175, 21. 




7. outwf] 
). nopiofidf 
m. Theon {BFVp}; cfr. ad prop. " 




ELEMENTORUM LIBER V. 26 

nam iisdem comparatis similiter demonstrabimus^ 
esse ^ = E. et alia quaeuis magnitudo est Z. itaque 
si Z magnitudinem ^ superat, etiam magnitudinem 
E superat^ et si aequalis^ aequalis est^ et si minor 
minor. et Z magnitudinis F multiplex est^ et ^, E 
magnitudinum A^ B aliae quaeuis aeque multiplices. 
quare T : -^ = F : B [def. 5]. 

Ergo aequalia ad idem eandem habent rationem 
et idem ad aequalia. 

Corollarium. 
Hinc manifestum est, si magnitudines proportio- 
nales sint, easdem e contrario proportionales esse.^) — 
quod erat demonstrandum. 

vm. 

Ex inaequalibus magnitudinibus maior ad idem 
maiorem rationem habet quam minor; et idem ad 
minorem maiorem rationem babet quam ad maiorem. 

Sint inaequales magnitudines AB, F, et maior sit 
AB^ alia autem quaeuis magnitudo sit z/. dico^ esse 
AB:A>r:AetA:r>A: AB. 

Nam quoniam AB> F, ponatur BE = R itaque 
minor magnitudinum AE^ EB multiplicata aliquando 



1) Quia et A : r = B : r ei r : A =^ r : B. ceterum hoc 
corollarium recte hic coUocatur in P; nam si post prop. IV 
fuisset, ubi Theon id posuit, alteram partem demonstrationis 
p. 22, 24 eq. superuacuam futuram fuisse, acute obseruauit 
Augustus n p. 331. om. Campanus. 

18. (jLsitov] x6 fjisC^ov P. 19. AB] P, Fm. 1, V m. 1; 

ABtov r Bp, F m. 2, V m. 2. itvzs Vp. 20. to z/;[(prius) 
TO in spatio 4 litt. qp. 23. AB'] B in ras. p. irai] to 9 
(non F). 24. t6] (prius) tw tp (non F). 




I 



26 ZTOixEiriN *'. 

xlaeia^ofievov eUtat. noTh zov ii (let^ov. lezm »( 
rfpoi' ro ^E slarzov xov EB, xai nmoki.aTtlaaiatS&a 
th AE, xal ^ffrro avtav aoklaakaeiov to ZH [iBt^ov 
ov tov ^, xal oeajcXiieiov ieti to ZHtov AE, toeav- 
5 xa%laaiov yiyovdza xal t6 nhv H& zov EB zo Sh 
K Toi) F' xal EiXrjgtQ-ai tou ^ SuiXaeiov fiiv to 
TptwAftiJiov ds to iW, xal a^ijg ivl %l8iov, eag av 
kajifiavo^evov JtoXXa^irldaLov (liv yivijtai zov ^, «pi 
tag 6e (letlov xov K. slX^^ip&oi, xal ieta xo 

10 zetQaTtXaeLOv fisv row -4, TtQatius Si fist^ov zov 
'Ensl ovv t6 K tov N nQazag dazlv SXaztov, 
K aga- roiJ M ovx iaziv IXazrov, xal iml iedi 
ietl noXXanldeiov ro ZH tov AE xal z6 H& ri 
EB, iadxts «p« ^ffti noXXa^Xdeiov zo ZH zov A. 
'-> xttl ro Z& zov AB. iedxis Se ieti noXXaaXdeiov 
ro ZH tov AE xal th K zov F' iadxig apa iatl 
noXXanXdatov ro Z& row AB xal t6 K tov F. tu 
Z®, K «Qa Tcov AB, F iodxi^ iatl noXXanXaOta. 
ndXtv, inel ladxig iazX noXXanXdaiov z6 H& tov 

EO EB xal zo K zqv T, taov 6e z6 EB z^ F, fffov 
aga xal tb H® rra K. ro di K tow M qvx iezi 
eXazzov ovd' apa tb H& zov M eXattov iarit 
p.etlov Se t6 ZH tov jd' aXov aga ro Z® ffwaj 
qiotiQav zav z/, M (let^ov iettv. aXXa ewajiiporEQtt 

26 ztt .d, M ta N ieziv taa, ineiS^nEg to M tov ^ 

1. mz£\ mg. F. 3, j4£1 F; ATS, %atq otr to isvo^vvt 
(isijov y^vJiTot lov a Theoa CBF''''?; ia F o^ corr. ei &v\ 
yivoftevov V, F m. 2), 5. lo 5s] «ai ro Bp. G, foC] (alt.) 
idv n (non P); to F, corr. m. 2. 7. ^iitrov] V m. 1; nXfCwi 
BFpjT, V m. 2. ffiv] ov P. 13. H0] S7f Bp et FV in 
raa, m. 2. tou] postea ioBert, F. 14. Antc ZH raa. 1 

litt. F. 16. Z0] Z in raB. m, 2 V. AS\ A in raa, m. 3 V. 
19, ^OTi"* F. 20, EB] AB F. xS] (alt.) corr. ex td m. « P. 



ov 



1 



ELEMENTORUM LIBER \ 



27 



laior erit magnitudino ^ [def. 4]. sit prius AE<iEB, 
et multiplicetur jIE, et sit mul- 
tiplex eiua ZH maior magnitu- 
dine ^, et quoties multipiex est 
., gZi/ magnitudinis AE, toties 
multiplex fiat H& magnitudinis 
EB et K magnitudinis F, et su- 
matur ^ = 2 ^, M = 3 ^, et 
deinceps multiplicea per unum 
creacenteB, douec sumpta mag- 
nitudo multiplex fiat magaitu- 
diuis z^ et prima maior magnitudiue K. sumatur, et 
sit N, quadruples maguitudinis z/ et prima maior 
msgmtudine K. 

iara quoniam K magnitudine iV prima minor est, 
K magnitudine M minor nou est. et quoniam ZH 
magnitudinia AE si H& magnitudinia EB aeque multi- 
plex est, erit ZH magnitudinia AE et Z& magui- 
tudinis j4B aeque multiplex [prop. I]. uerum ZH 
magoitudiuis AE et K magnitudinia V aeque multiplez 

keBt. itaque Z& magnitudiuia AB et K magnitudinis 
r aeque multiplex est. quare Z&, K magnitudinum 
^B, r aeque multiplicea aunt. rursus quoniam H& 
pnagnitudinis EB et K magnitudinis F aeque multiplex 
Mt, et EB = r, erit etiam H& = K. uerum K 
magnitudine M minor non est. itaque ne H& qui- 
dem magnitudine M minor est. aed Zff > ^. 
Z® > ^ + M. sed z/ + M = iV, quoniam M= 3 ^ 



ov3l comp. 
!M. «] in rtw. 1 



^Bs3. ov3i 




28 STOIXEIiiN f". 

TffinXdeiov iextv, evvafttpottpa 8i ra M, ^ tw ^ 
iett TEtQec%Xa6i,a . ieri. dl xal to N tov /i terpa- 
nXaSiov oi-vHfiqpotfpK cpor th M, ^ ta N tea ietiv. 
dkXic ro Z0 rmv M, /i ^Eltfiv iaziv to Z0 apa 
5 tov N vasffixif to Sl K tow A' ov% vnBQi%ft. Mci 
ieti ta (tJv Z&, K rav AB, F ladxig jtolXaalixaia, 
ro di iV rou ^ aAAo, o itvxtv, aolXaxXdeiov to 
AB aga npog ro ^ (uCtflVtt Xoyov i%ei. ^jctp to F 
jEpog t6 -^. 
10 Aiya 5jj, oti wtl to ^ Jipo^ to F (iBltfiva kof^ 

f^XEt T(Sfp TO jd KpOff to ^B. 

TtSv yap awTrov xaraexivaeS^ivtmv ofioto; tfeP 

loftfi', OTt t6 (ifv iV Tou K vnsQiiEi, t6 tfi N rov 
Z© ovj; vntpixst. xai iati to fi\v N xov jj jcoXXa- 

16 jtXdaiov, td 6h Z0, K tmv AB, F aXXa, a izvxsv^ 
iedxtg aoXXanAdeia' t6 ^ dpa XQog ro f" (lei^ot 
Koyov i%et ijmQ ro z/ rcpog r6 ^S, 

'AXld di] to -4jE To{i EB ikC^ov ietto. t6 i^ 
iXartov t6 £5 reoAAn;rAffB(«5''f**'''"' *6tat jtote tou 

20 ^ (if r^ov. itSJCoXXanlaetdB&a , xal ietio ro H& 
stoXXaaXdatov {liv tov EB, fitt^ov dl rov .J' xal 
oaaxXdaiov iett zb H& roiJ EB, zoaavTanXdaiov 
ytyovita xal lo (ihv ZH tou AE, ro 6i K rou F. , 
oiioiag drj dti^ofiev, ort tk Z0, X rtov .^S, f" iadxt 

25 ^iiTi itoAAanAaoitc ' xal elX^^cp&m o^otrag t6 iV scoAili 
nXdaiav (itv rou ^, ffpoitmg de ^ftgov rou Zif" 

1. ^oiii'] B, comp. p; om. F; laxt PV. Si] c 

m. rec. B. M, J] J, M P. 2. 10] corr. ei xoi 

3. IffTti' laa FT. *. 1.0»] i(5 K V. JW, *!] .d, jHE. - 

imi BV. Z0] ZE 9. 7. fti^ie ^ (non P) Vp. 8. ciqm'' ' 
m. 2 F. 12. Sr} Ssi^ioiiiv P. 13. jiiv] m. 2 F. 
corr, ex ov* m. 2 P. 16. ru] 10 Fp. Z9, K] iitt. », KM 



lor^M 

{ xov 
oAAa- 



ELEMENTORUM LEBER V. 29 

etM+ ^«=4^ etiNr=4z/; itaque M+jd^N. 
sed Z ® > M + z/. itaque Z & magnitudinem N superat. 
K autem magnitudinem N non superat et Z@, K 
magnitudinum AB, F aeque multiplices sunt^ N autem 
magnitudinis ^ alia quaeuis multiplex. itaque AB 
:^>r:^[def. 7]. 

dico igitur^ esse etiam jd : F > ^ : AB. nam 
iisdem comparatis similiter demonstrabimus, N magni- 
tudinem K superare^ Z® autem magnitudinem non 
superare. et N magnitudinis jd multiplex est, Z&, K 
autem magnitudinum AB, F aliae quaeuis aeque mul- 
tiplices. itaque A : r> A : AB [def. 7]. 

E iam uero sit ^B > EB. ita- 

^ ' ' ' ^ que minor magnitudo EB multi- 

^' ' „ plicata aliquando magnitudine 

Zi 1 1 — I — \9 A maior erit [def. 4]. multipli- 

iti— i-H cetur, et sit H® magnitudinis 

^i — I EB multiplex et magnitudine 

^, — I — , z/ maior. et quoties multiplex 

M\ — I 1 — I est H& magnitudinis EB^ toties 

iVi — 1 — 1 — I — I multiplex fiat ZH magnitudinis 

AEetK magnitudinis F. iam similiter demonstrabimus, 
ZSy K magnitudinum AB^F aeque multiplices esse. et 
similiter sumatur N magnitudinis ^ multiplex et 
prima maior magnitudine ZH, quare rursus ZH 



ras. m. 2 V. a] m. 2 F. ^ 18. xov EB (iBitov ^ctm] P; 

(ist^ov ^atm Tov EB BVp; tov EB m. 1 F, seq. (ieitov tctto 
Tov EB (p. to ^Tj ^lattov t6 EB] noHanXa ip. 20. 
nsnoXXanXuaidad^io] post ns- ras. 2 litt. F. 23. fjkiv'] ip in 
spatio plurinm litt. to] in ras m. 1 p. 24. Ta] to q> 
(non F). 




30 ETOIXEiaN t'. 

foets XaKiv zo ZH tov M ovx ietiv tlaoaov. 
^ov 6i ro H& rov A' oXav aga ro Z@ tav ^, M, 
tovttuzi. tov N, VTtBSfixet.. to Sb K tov N ovx vK£p^2**j 
iatiS^^niQ xal ro ZH (isi^ov ov roti H&, tovtiati 

5 Tou K, roi; JV oiij; vnB^iisi. xoti mtnxvtaiq xava- 

xoi.ovS-ovvTBg tois i^avm nsgaivofiBV tijv dii6d6i-^i.v. 

Tav UQa uviaav jieye&i^v to ftei^ov Jtpos to «vto 

[leitova Aoyoi' ix^i ^asQ to iXattov xal to avro 

jrpoe to eA.attoi' [isi^ova loyov B%Bt iqnB^ JTpog T^ 

10 ^jrjoi'" ojtBQ idti SsL^ai. 



Tu jTpog to avth xov aitov ixovra Xoy 

tsa alX^Xois ieziv xal x^og k to ocura 

avtov i%bi loyov, ixetva fffa istiv. 
I 'Exita yap ixatsQOV tav A, B wpos tit T xy 

avtov AoVo"' Xiym, OTt iSov i6tX to A ta B. 
Ei yaff fiij, ovx av ixazcgov tiav A, B «pog xo 

Tov avtbv Btxs XoyoV i%Ei Si' toov apa iotl to 

Tp B. 
t 'E%stat Sij xdXiv to F Kpog exdtSQOV tmv A, B 

tbv avtov loyov Xiyto, oTt taov ietl to A tj 

Ei yaif ^)J, ovx av t6 F npos ixdzBQOv tav A, H 

tbv avtbv eI%£ kayov i%st 6i' teov aqa itSxX xo 

Tco B. 
> Ta agu Mpog to avtb zbv avzbv exovta ioyt 

tea dXk^Xoig iativ xal npog a to avtb zbv avzot 

ixsi Xoyov, ixsiva Caa iativ ojzsq sSei Setiai, 



,B 

I 



1. ovK iativ tlaaeav] (iq tlaaaov tlvai P. rlOTTOV Fp. 
, nnv] Tov Bp. 3. toviiaiiv P. ovi iijifp^jtt] ijttet- 
ov3tt(Las V. 4, ineidrjjcgq -— d-.vnefi^ti^ mg. m. 1 F. 



ELEMENTORUM LIBER V. 31 

magnitudine M minor non est. et H® > z/. itaque 
Z0 > ^ + iVf, h. e. Z@ > N. K autem magni- 
tudinem N non superat^ quoniam ZH^ quae maior 
est magnitudine H®^ h. e. maior magnitudine K^ 
magnitudinem N non superai et eodem modo 
superipra sequentes demonstrationem conficimus. 

Ergo ex inaequalibus magnitudinibus maior ad 
idem maiorem rationem habet quam minor; et idem 
ad minorem maiorem rationem babet quam ad maiorem; 
quod erat demonstrandum. 

IX. 

Quae ad idem eandem habent rationem, inter se 
aequalia sunt; et ad quae idem eandem habet rationem, 
ea aequalia sunt. 
A B Sitenim^:r=B:r. dico, 

r esse A = B. 

nam si minus, non esset ^ : JT = JB : JT [prop. VIII]. 
at est. itaque A = B, 

iam rursus sit F: A = F: B. dico, esse A = B. 

nam si minus, non esset F: A = F :B [prop. VIII]. 
at est. itaque A = B, 

Ergo quae ad idem eandem habent rationem, 
inter se aequalia sunt; et ad quae idem eandem habet 
rationem, ea aequalia sunt; quod erat demonstrandum. 

4. ov] corr. ex mv m. 2 P. 6. tov] (prius) P; rd BFVp. 

%UTav.oXov^QvvzBg] bis P; corr. m. 2. 6. awodfijtv] post 
dnO' spatium 1 litt., in quo m. 2 inser. ds F. 8. xo iXaxxov 
— 9: rinBq] mg. m. 1 P. 13. hxCv] F; comp. p; kaxCVBN, 

a\ euan. F. 14. Y,a%Biva V. 17. f*^] itsiiov 9. 18. 

slxej in ras. P9, slxsv B. I^^O Hv 9* 23. «Ijfc] in ras. P; 
ixsi B; ^xV F. iaxiv F. 26. iaxCv] comp. Fp; iaxC PBV. 
27. %dHSiva V. 




Tav TtQog to avto Xoyov ixovzav tb ftBt 
^ovce Adyov exov ixBtvo ^il^ov iertv xffoq 
Si To avto fisi^ova Xoyov Ij;*'? ixeEvo Ikatri 
5 iettv. 

'Exitco yiiQ ro A n^og x6 F (ititova Aoyov ^«e( 

To B Jtpog ro F' laym, oti (let^ov ieti to A rou B. 

Ei ya(f /iij, ^toi fffov tffti tb A ta B ii llao- 

aov. ttsov fikv ovv ovx iffrt tb A tm B' ixutfQOV 

10 yap «v TiDv v^, B ffpos t6 r" Toi' avtbv slxe Aoyov. 

ovx Ix^i di- ovx apa tsov iotl tb A ta B. ovSe 

jiiiv iXaaSov ieti tb A rot) B- tb A j-Kp av wpoff 

tb r iXaeCova Xoyov elxsv ^neg tb B ^hqos to f, 

ovx i'j;s(. Sd' ovx Sga IXaOSov isti to A tou B. 

15 iSeix&y} S\ ovS\ teov fi£i%ov a^a iorl tb A tou B. 

'Exeta Sfi itaXiv zb F itQog tb B (isi^ova Xoyov 

^MQ tb r Jipog To A' Xiyoi, ozt eXaGSov ioti. t{ 

B tov A. 

EC yuQ fi)j, ^Tot leov iatlv 7} ^Er^ov. fffov 
20 ovv ouK iatL tb B zm A' tb r yag av apog ^koi 
Qov tav A, B tbv avtbv elxe Xayov. ovx Sx^t S('- 
ovx aQBL taov iazl 10 A za B. ovSi (Itjv (ist^i^ 
iazt tb B low A' tb P yaQ av «pos t6 B ikaecovtt 
Xoyov elxsv ijTteQ Ttgbg zb A. ovx £;i;£i Sd' ovx apc 
2& ftst^ov iatt tb B zov A. iSeix^n Se, oti ovSi teov 
IXattov Bpa iazl 16 B tou A. 

Tav &Qa Jipog tb avxo Xoyov ixovtmv tb ^M^oiUf 
3. ro tov tigiiovu V, 3, lon*] P, 



.Tp] t 



A^ 



B]i 



] PV; 



i ioTi BPV. 
8. iari] <p. l&civ F. 



i 



ELEMENTOBUM XJBEB V. 33 

X. 

Eorum^ quae ad idem rationem habent, quod 
maiorem habet rationem, id maius est; et ad quod 
idem maiorem babet rationem, id minus est. 
A\ — ^ 1 JBi 1 Sit enim A : F> B : r. dico, 

r\ 1 esse A> B. 

nam si minus, aut A = B^ aut AKB. uerum 
non est A = B] tum enim esset A : F = B : F 
[prop. VII]. at non est. quare non est A = B. 
neque uero A < B] tum enim esset A : F < B : F 
fprop. Vin]. at non est. quare non est A < B. 
sed demonstratum est, idem ne aequale quidem esse. 
itaque A> B. 

sit rursus F: B> F: A, dico, esse B <A. 

nam si minus, aut B = A aut B > A. uerum 
non est B = A] tum enim esset F : A = F : B 
[prop. Vn]. at non esi itaque non est A = B. 
neque uero B> A^ tum enim esset F : B < F: A 
[prop. Vin]. at non est. quare non est B> A. sed 
demonstratum est^ idem ne aequale quidem esse. 
itaque B <A. 

Ergo eorum^ quae ad idem rationem habent^ quod 

10. slxs^^xsi B; F, corr. m. 2. ^ 12. lAaTTOv F. 13. tov 
ildaaova v; ituttova F. bIxb Xoyov P. 14. ^AatTOtr F. 

iati] m. 2 F. 15. d^ OTt V. Post B repetantnr in F : 

ids^X^ dh ovSh taov (isl^ov aga to A tov B, 17. ^Xattav 
F. 20. A] in ras. m. 1 B. yaQ] insuper comp. add. m. 

2 V; «^tt B. 21. slxs] 9; slxsv PB; ^x^i F. 22. iat^ 

iazLv P; comp. F addito iati y. to] to5 V.. t&] to V. 

23. iativ P. To] (prius) tov V, corr. m. 1. iXattova F. 

25. ov8' (p {non F), -s in ras. m. 1 B. 26. iXattov] 9, seq. 
oy m. 1; ^Xaaaov P. 27. to] (alt.) om. 9. (usliova] 9, 

seq. ova m. 1, eras. 

Euclides, edd. Heiberg et Menga. II. 8 



ETOIXEIJiN ( 



Xoyov ijiov (ist^ov ietiv xal stQog o to avro fiei^ova 
ioyov %c(, dxtlvo ikarxov iativ' oncff idtt dtt^i. 



TO 



0[ T^ avzp Xoya ot avtol xal aXk^^Xoi 
6 tttlXv ot avxoC 

Esrcaaav yaff 6g [liv lo yi ^HQog ro B, ovtatg 
To r ffpog ro z/, mg Si ro F Tcpog to ^, ovxas t6 
£ sTpos To Z' Xtya, ou ^«tIv tog ro v^ Jrpog 
orTOJS t6 E JTpO? to Z. 

10 ElXriifi&a yftp T(5v ^, F, E ieaxtg xoXXaxXaei 
T« H, @, K, tmv ds B, J, Z aXXa, a srvxsv, iad: 
noXXanXdaia za A,M,N. 

Kal iati ietiv mg t6 j1 nrpos ro B, ovzets 
r Jrpog ro ^, xal eHkTjmtai tmv fiiv A^ F lam 

16 noXXaitXuaia Ta H, ®, tav 61 B, d aXXa, a hvxEV, 
ladxis ^oXXaxldtJia rd A, M, d apa vm(fixH ro 
H Tou A, vatQEx^i, xal ro & ToiJ M, xal ii ieov 
isrlv, laov, xal st iXXstxst, iXXeinet. ndXiv, exei 
ieztv ras '6 r JtQog ro jd, ovrias ro E ;rp6s t6 Z, 

20 xal etXrintttt rmv F, E iedxis noXXanXdaia zd @, K, 
zav &\ jd , Z aXXa, d itv%ev, laaxig TtoXXamXaata 
rd M, N, d dpa vxe^ix^i r6 & rou M, vnsgix^'' 
xal To K Tou N, xal eI taov, toov, xal tt iXarrov, 
IXartov. dXXd ti vnsgstxs to & tou M, vttsQeSx^ 

25 xal rb H rou A, xal tt teov, teov, xal st iXazrov, 

1. ^OTi»'] B, oomp. pi 1«. PFV. B. Haaoo* PBVp. 

4. liym] F m. 1, F, Vm. Ij idyoi Bp, Pm. 2, g., Vm. S. 

6. ovtm P. 11. ^, Z] Z, J F. «J ecoiT, F. 13. t«] 
la H, 0, K td P, corr. m. 1. 14. /leV] m, 2PV. F] in 
rae. m. 2 P. 16. JT) in ras. m. 1 p. St] om. 9. B, .dl 
H, ,^ 9 (non F). SXXa lcaytii noi.luvXeiata S ^ivxe V. bJ 



i 



^m ELEMENTORUM LIBER V. 35 

maiorem habet rationem, id maius est; et ad qnod 
idem maiorem habet rationem, id mistis est; quod 
erat demoQstraiidum. 

% ...'''■ 

^R- Quae eidem rationi aequales suut rationes, etiani 

^Bjnter se aequales auBt. 

H Sit enim A : B = r : ^etr:z/ = £:Z. dico, 

H^i 1 ri 1 Ei— I esse J:B = E:Z. 

^Kji — i ji — I Zi-i aumantur enim 

^^Ki 1 1 Si 1 1 Ai— 1— I magnitudinum ^, 

^^ral — 1 — I — 1 Mi — ! — I— ( iVi-i-i-i r, Eaeque multi- 
"^^lices H, &, K et magnitudiimm B, J, Z aliae quaeuis 
aeque multiplices A, M, N. 

et quoniam A : B = V : jd, et sumptae sunt magui- 
todinum A, raeque multiplices H, © et magDitndinum 
B, iJ aliae quaeuis aeque multiplices A, M, si H 
magnitudinem A superat, etiam magnitudinem M 
saperat, et si aequalis, aequalis est, et si minor, 
[def, 5]. nirsus quoDiam V: /i = E: Z, et 
siimptae sunt magnitudinum F, E aeque multiplices 
®, K et magnitudinum A, Z aliae quaeuia aeque multi- 
plices M, N, si ® magnitudinem M superat, etiam K 
jDagnitudinem N superat, et si aequalia, aequalis est, 
si minor, minor [def. 5]. sed si ® magnitudinem 
saperabat, etiam H magnitudinem A superabat^), 

l) Imperfectum recte eb habet; refertur enim ad ea, qaae 
lam lin. 16 sq, dicta Buot; cfr. p. 60, IS. 

m. 2 F. 17, H] in ras m. 2 V, 20. tmv niv P. K, « p, 
81. J] Jt. J F, sed corr. S] m. 3 F. 22, tov] m. 2 V. 

2*. al!.a el — 26: IXarTov Jalt.)] mg. m, 2 FV (dXl\ 24, 
vntpffZ^l iniQgtzey corr. es vatQfxii m. 1 P', vmQixti BFVp. 
4)U(e('{iJ p; vjtigtl^xfv FB; vTiii/fxti FV, 



mag 
^^sape 

plii 
jna 



36 



ETOIXEIflN e 




ilattov mara xal bI vxtff^x^i z6 H rotr Af hn^^ 
kkX t6 K tov N, kkI bC teov, taov, xal tC SkaxzoV, 
Skattov. xttC isri ra (lev H, K tav A, E CsaxiB 
nolf.a3tXK0ia, ta d} A, N tav B, Z alka, a hvxevA 
6 Ceaxis nolXaaldaia- ietiv «qu mg xo A a(/os to > 
ovras ro E XQog to Z. 

01 aqa r^ avx^ Xoya ol avrol xal dXX^loig eCsliK, 
oC aitoC- Sme idti Set^ai. 

'f- 

10 'Eav Ji bxoaaovv ^Byi^T\ dvdXoyov, iarccm 
d)s av Ttov rjyovfiivav jiqob 'iv tmv ijtofidvmV^ 
ovTcas djcavta ta Jiyov(iEva ngog aaavta : 
iaofifva. 

"Eataaav osoCaovv fieyi&n dvdhoyov ta A, B, ] 

15 A, E, Z, tos ro A hqos t6 B, owrtBs ro F mpog zo . 
xal t6 E Jtfos t6 Z' Xsyco, ort iazlv ag zb A fl^oq 
t6 B, ovteas ta A, f, E xgbs ta B, A, Z. 

ECli^^&a yap ttav fiiv A, F, E Cadxiq JtoXXanXa-- 
aia ta H, &, K, tmv Se B, A, Z SXka, a hvxev, i 

20 ladxis noXkanXdaia ta A, M, N. 

Kal ijcsC iettv wg ro A «pog zb B, ovtms «o 
r JtQos tb A, xai r6 E xqos to Z, xal stXtjOzai 
tmv (isv A, r, E Cadxis xoXXajtldeia td H, &, K 
tmv Se B, A, Z aXXa, a hvxev, iedxis xoXlaitXdaia 

85 TK A, M, N, &C djfa vasQBxtt tb H zov A, vXEffixet 
xal rb & Tor M, xal ro K roi! N, xal bI reov, feov. 



XII. Eutociua in Archim. ni p. 186, 26. 



2. flaatiov, fXasaov V. 4. Z] J P. a] supra F. 7. ^toyvln 
Pj lifoi BFVp. le. huv] om. F. 17. tu] to F, " 



ELEMENTORUM LIBEIl V. 37 

et si aequalis, aequalis eat, efc si minor, minor. 
quare, ai H magnitudinem A siiperat, etiam K mag- 
nitudinem JV superat, et ai aequalis, aequalis est, 
et si minor, minor. et H, K magnitudinum A, E 
aeque multiplices sunt, et jI, N magnitudinum B, Z 
aliae quaeuis aeque multiplicea ; erit igitur A : B = E: Z 
[def. 5]. 

Ergo quae eidem ratioiii aeqnales sunt rationes, 
etiam inter se aequales aunt; quod erat demonstrandum. 

sn. 

Si quotlibet magnitudinea proportiouales sunt, erit 
ut una praecedentium ad unam sequeutium, ita omues 
praecedentes ad omnes sequentes. 

Siut quotlibet magmtudines proportiouales j^, B, 
r, J, E, Z, ita ut ait ^ : B = r: ^ = E : Z. dico, esse 
A:B = A + r-\-E:B-{-^ + Z. 

sumantuj enim magnitudinum A, F, E aeque multi- 



plices H, @, K et magnitudlnum B, ^, Z aliae quaeuis 
aeque multiplices A, M, N. et quoniam est A : 
= r:A = E:Z, et sumptae sunt magnitiidinum A, F, 
E aeque multiplices H, 6>, K et maguitudinum B, A, Z 
aliae quaeuis aeque multiplices A, M, N, si H magni- 
tuiUnem A superat, etiam ® magnitudiuem M superat 

10 P, aed coti. fij poetea ineert. F. 19. a] m. 2 F. 
(«■»] om. Bp. 24. o] m. 2 P, 23. H] in ras. F. 



38 



ETOIKEIHN t 




xttl ti ^AoTcoi' , i?.attov. mstB naX ei vniQsj^si i 
H tov A, vitEQtx^i xal ta H, &, K tav A, M, N, 
Xttl bI taoVf taa, xal bI Ikaxzov, ilartova. xaC iari 
rh fih/ H xal ta H, &, K toO A xal twv ^, F, E 

6 ladxig xoi.Xttald0ia, ixtid^itB^ idv ij oitoaaovv lisyd&Tj 
OTtoaavovv (leyB^ap tOav ro jtXri9os BxaOtov ixdatov 
ladxig aoKlaTtXdaiov, oaaxXdaiov iariv 'iv rmv (isyE- 
9mv Bvoq, roaavranXdaia iarai xal r« itdvra tinv 
jidvtav. ilitt ra avrd Sij xal to A xal %a A, M, N - 

tov B xal tmv B, ^, Z (adxig iotl JtoXXttxXdeia^ 
iariv apa mg ro A X(/ag t6 B, ovras ra A, T, . 
«pos ril B, ^, Z. 

'Edv aga fj onoaaoiJv jiByi&i} dvdloyov, iaral^ 
ms h/ Twv Tjyovfiivav irpog ^v rmv iaofiivav, ovttoi 
l& anavra ri riyovfiBva ropos dnavttt zd inofiBva' 



'Eav itQiatov nQos Sbv 

Aoyo!' xat iQirov jrpog 
itpog T^raprov jifi^ora X 
Ttpos i 



■EQOV tov avzov SxV 
itttQtov, rffitov 8% 
ayov ixti ^ ni^ntov 
I, Kttl npturov irpos dBvrBQov ^Etgovixl 
Xoyov £'§£( ^ niftnrov «poj exzov. I 

IlQmtov yitQ to A jrpog SBVtBQOv tb B row ccvrov^ 
ixira loyov xal Tfiirov to f jrpog titcefftov ro ^, 
25 zQitov Sh ro r ngos tiraptov ro z/ iiBi^ova Xoyov 
i^iza ^ nB(intov rd E irpos Jxrov t6 Z. Xiya, ori 
xttX nQtozov ro A Ttpog SevtsQov ro B (isitova Adyoi 

i|fl ^KBQ lcillTCZOV TO E ItQOS BXZOV TO Z, 

1. «aaifov rHarffin)»' V. 2, rd] to P. tiv] ro5 

3, Caa] teov PBp. aoflffov iXacsov P; flarTov aoTTDv I 
6. liv] ttv P. G. ffftovl fijov BF. 7. nollojiliiina 

10. tov B] litt. B e cort, F. imC] iotai p, 11, td] 



IT( 



ELEIMENTOEUM LIBEE V, 39 

et K magnitudinein N, et si aequalis, aequalis eat, 
et 81 minor, niinor [def. 5]. quare, si H magni- 
tudinem A auperat, etiam H -\- & -\- K magoitudines 
A-\- M-\- N superant, et si aequalis, aequales aunt, et si 
miaor, minorea. iam H magoitudinis AetH-\-®-{-K 
magnitudinum A -{- F -^- E aeque multipliees suut, 
qooniam si datae suut quotuis maguitudines quotuis 
magnitudinum numero aequalium aingulae singularum 
aeque multiplices, quoties multiplex eat uua magni- 
tudo unius, toties etiam omnes omnium erunt multi- 
plices [prop. I]. eadem de cauaa etiam A maguitudinia 
B ei A -\- M -\- N magnitudinum B ■{- .J -\- Z aeque 
mnltiplicea suut. itaque 

A:B=A-{-r-\-E:B-{-J-\-Z [def. 5]. 
Ergo si quotlibet magnitudines proportionalea 
sunt, erit ut una praecedentium ad unam sequentium, 
ita omnes praecedentes ad omnes sequentes; quod 
erat dembnstrandum. 

xm. 

Si prima ad secundam et tertia ad quartam eandem 
rationem babet, tertia autem ad quartam maiorem ra- 
tionera habet quam quinta ad aextam, etiam prima 
ad aecundam maiorera rationem Labebit quam quinta 
ad seztam. 



Bi— . 


ji 1 




Wi — 1 — i 


— 1 ^i- 






_i 


Sit enim A:B = 


r: 


;z/et r; 


■.^> E: 


Z. 


dico. 


eaae 


etiam A 


:B> E:Z. 















FV. H. idl to F. 16, Snavxa] (alt.) jtdvra P. SO, 

n] P; TJnte BFVp. 23, 17] P; qnfj BFVp. 23, ftiv yag 
P. xdv B¥. 26, ^] P, Fm. 1; ^Ttsp BVp, Fm, 3. 28. E|*t] 
Ixft P. f,ntf 10 E jt^oe To Z P, 



40 STOIXEIflN /. 

'Entl yitg eari ztvtc rav p,\v F, E ieaxig xoi 
jikiieiK, Ttoi/ fsl z/, Z aXka, a hvxiv, inaxis itoXh 
TcAaaia, xttl to fiiv tov F JtoXf.a%i.d0tov tav tot' ^ 
xoKkaitlasiov {msqb%€i, ro d^ rou E 7to!iKaTti.ttmov 

B tov toiJ Z TtolkajikttCiov ovx vnsQfx^t, £fXijip9iOf 
jml iotat Ttov filv F, E Itfaxis aoAXaalaata r« H, 8, 
Ttow Sl jd, Z ttXXa, a hvxei', ladxis noXXajiXaattt 
za K, A, (oBTe to ^\v H tov K vat^exeiv, ro Sh 8 
roi» .<4 ft^ vjtiQextiv xal oeaxXdaiov }iiv ioti tii 

H rou f, TO«aiiT*wrX«eioi' sOto} xal tt J 
offa^rAtJecof S\ t6 K roii z/, toaavtanXdatov ieti 
xaX t6 iV Tov B. 

Kttl ixti iattv rag lo A JEpog lo B, ovttng ro 
wpog t6 ^, xal stXviTtxat rrov /tiv A, V iadxig sco^^i 

B nXdattt ta M, H, tSv dh B, ^ aXXa, a hvxsv, let 
xig TtoXXttJiXdaia r« N, K^ sl UQtt vjtEqix^t ro 
roii A'^, vmtQixst xal ro H tov K, xal tl teov, fffOfJ 
xal eI ilattov, IXttztov. vnEQSxit Sh to H zov K' 
vXEffix^i. Kpa xal tb M tov N. to dh @ tov A ov% 

{ntEpEXEf «ai iett za ftiv M, & tav A, E ledxis 
jtoXXaaXdaia, ra Sh N, A irov B, Z aXXa, a ItvxBi 
ladxig JtoXXanXdeia- to aga A npo^ td B itei^o; 
Xoyov ^x^t ijjTfp ro E jrpog ro Z. 

'Edv aQtt jrptoTow nifot; SevttQOv rov atitov Ij 



L 



1. Post yiig add. Theon: ro F ngbs lo J fM/£oM! Idyori 
f;(£( ^ittQ To E mpoe lo Z (BFVp); om. P. 2. luv aj ^, t 
— aolXojtldintt] mg. m. 1 P. 3. ro] corr. ei rn m. 1 V, 

xov] (alt.) postea ijiBert, m. 3 F. 7. o] supra _F. 8. 

Ante ineefjei»' raa. 2 Utt.. V. 9. fuj] P; ou fin Fj 0»« 

BVp, 15. of] snpra m. 2 F. 20. rd] corr. es ro jn. 1 "V^ 

-^lin ras. P. 21. tic 8e — 22: boUkiiIqiho] 
to m/a A icpos 10 £] iu rsG. m. 2 F, seq. ueatig. 12 li< 

24, hn V. 






liii^ 




ELEMENTORUM LBBER V. 41 

nam quoniam sont quaedam^) magnitudinum F, £ 

^ aeque multipliceS; magnitudi- 

„ num autem z/, Z aliae quaeuis 

^ aeque multipliceS; et multiplex 

aI J-K_, -nagnitudinis T multiplicem 

magnitudinis ^ superat^ mul- 
tiplex autem magnitudinis E multiplicem magnitudinis 
Z non superat [def. 7]^ sumantur^ et sint magnitudinum 
r*, E aeque multiplices Hy @^ magnitudinum autem z/, Z 
aliae quaeuis aeque multiplices K, A^ ita ut H magnitu- 
dinem K superet, & autem magnitudinem A non superet. 
et quoties multiplex est H magnitudinis r*, toties 
multiplex sit M magnitudinis A^ quoties autem multi- 
plex est K magnitudinis dy toties multiplex sit N 
magnitudinis B. et quoniam est ^ : £ « F: z/, et 
sumptae sunt magnitudinum A^ F aeque multiplices 
Af, H, magnitudinum autem B^ A aliae quaeuis aeque 
multiplices N^ K, si M magnitudinem N superat, 
etiam H magnitudinem K superat^ et si aequalis, 
aequalis est, et si minor, minor [def. 5]. uerum H 
magnitudinem K superat; quare etiam M magnitu- 
dinem N superat. autem magnitudinem A non 
superat. et M, (S> magnitudinum A^ E aeque multi- 
plices sunt, N, A autem magnitudinum By Z aliae 
quaeuis aeque multiplices.^) itaque 

A:B> E:Z. 

Ergo si prima ad secundam et tertia ad quar- 
tam eandem rationem habet, tertia autem ad quar- 

1) jLi^v et Si lin. 1 — 2 inusitate quidem posita siuit, 
neque tamen ita, ut ferri nequeant. 

2) Cfr. lin. 6—8 cum lin. 9 eq. 



42 ETOrXEIiiN t\ 

loyov xal rsfitov repog xitaQiov, tgitov 
Thagtov (iti^ova koyov ^^V V ''^^('■^'^ov iZQog ixzov, 
xttl n^azov ^gbs SivtiQov fiiC^ova koyov e^ti 
ztfiJttov HQog ixzov oneQ idti SBt^ai. 



■.ov Sh itffiiq 

ov, 

V 

1 



'Etcv agazov ^rpog StvtcQov tov ttvrov l^V 
Aoyor xal tgizov npo? tizaQzov, zo 8\ ^rpm- 
Eov zov tpizov (ist^ov ij, xal zo SsvTtpov ti 
ZEtaQtov (itt^ov lazai, xav isov, fffov, xt 

10 ikaztov, ilazzov. 

npmzov yaQ To A ntQog SsvrtQOv ro B tov o 
ror ix^zfo Xoyov xal ZQizov zo F repof zdtuQzov 
^, liti^ov Ss lara to ^ tou F' Xsyat, ori xal ro 
Tofi z/ [lEi^ov iativ. 

15 'jSitEl yap To ^ zov r (isttov ieziv, «klo di, 
£rv-(sv, \y.iys^os\ t6 B, ro A UQa ZQog to B fiti^ovt 
Xoyov ixti TjntQ ro F repog ro B. aq Si t6 A jiqos 
TO £, ouTOjg To -r" XQog to ^' xal ro P «pa 3[p6g 
To i^ /lE^^ova Xoyov sxtt ^ittQ t6 T ffpos to [B, 

80 Kpog tfi t6 avto ntitova Xoyov ixsi, ixetvo ilat 
aov iativ ikaaaov Squ t6 ^ rov B' aett fuCgcfj 
iati t6 B tou ^. 

'Ofioicas S^ dti^ofitv, ozt xav teov 7j z6 A za P, 
i'aov Sarai xal ro B ra ^, xav ikaOOov in th A 

25 rov r, ikaaaov ierat xal rd B rov ^. 

'Eav uQa aQatov TiQog StvrtQov zov avzov Ij 
loyov xal zQirov ttQog zizaQzov, z6 Si JtQiBzov t< 
Tp^rou fiBi^ov y, xal ro StmtQov rou tttdQzov fitt^t 
ietai, Xttv (aov, iaov, xav ikaztov, iXarzov 

80 ISsi SsttttL. 



1 



ELEMENTORUM LlBER V. 43 

tam maiorem rationem habet quam quinta ad sextam, 
etiam prima ad secundam maiorem rationem habebit 
quam quinta ad sextam; quod erat demonstrandum. 

XIV. 

Si prima ad secundam et tertia ad quartam eandem 
rationem habet, prima autem tertia maior est^ etiam 
secunda quarta maior erit; et si aequalis^ aequalis erit, 
et si minor, minor. 

A\ 1 T\ 1 Sit enim ^ : B = f: ^, et 

jBi 1 ^i 1 A>T, dico, esse etiam B > ^. 

nam quoniam est ^ > F, et alia quaeuis magni- 
tudo est B, erit A\E>r\B [prop. "Vlll]. uerum 
A\B = r\ A. quare etiam F : z/ > F : B. sed ad 
quod idem maiorem rationem habet; id minus est 
[prop. X]. itaque B> A. 

similiter demonstrabimus, si A = F^ esse etiam 
B = Ay et si A <ir, esse etiam J5 < z/. 

Ergo si prima ad secundam et tertia ad quar- 
tam eandem rationem habet, prima autem tertia 
maior est; etiam secunda quarta maior erit, et si 
aequalis^ aequalis erit, et si minor, minor; quod erat 
demonstrandum. 



2. x6 thaQTOv B. ^x^^ ^9* V^^Q V9. 8. i^nsQ Ytp, 
9. xavl xal av V. %av] xal &v Vg>, 13. A] d g>, 

16. ftsiiov iatt. to A tov F P. rol corr. ex rov V. xotil 
corr. ex rd V. 16. ^tvvs Vp. iisys^^oQ} om. P. 20, 0] 

m. 2 P. I^AarTov F. 21. httttov F. 23. |?] supra m. 1 F. 24. 
xay] xa^, supra scr. kdv m. 2 Y. ^Xattov F. 25. I^Zar- 
tov F. xa^J om. V. 26. nQmtovl -tov in ras. m. 2 V. 29. 
iXaaaov ^Xaaaov p. 



44 ITOrEEiBP »'- 

^ ro ^£ r<a* Z' ^«70. ori imiv m^ ro F x^o^ ro Z, 

ovro; xo AB x^g ro ^£. 

'£i:r£( ^ap I^omis ^ri MkdJtauuiowov xo AB tm 
r Tuti ro ^^ ror Z. cKkr cpc i^ir ir r^ -^JB fXr 
yi^tl l^ rp F, rotfcrrc auu ir rp -^i-B Ita ry Z. 

1^'' iitj^fi^o ro fi^r >^B £/^ ra ro F ite ra jf ^ fl^y 
€^B, ro d« ^^ d^ xa ro Z /te ra AK^KA^AE' 
i&tm df^ l6ov ro nlffios rav AUj HB^ BB r^ «1^- 
#i(^ ror AK. KA, AE. Moi isul loa iotl xa AHj 
HB^ BB uXkrikoiSj iori dl Moi ric AK^ KAj AE 

l^ t0a d?J,rii?.ocg, ioriv uga o^ ro AH xgog ro AK^ 
oikws ro HB xgog ro KA^ xal rb SB XQog ro AE. 
i&tuv UQu xal wg Zv rav f^yoviuvav XQog ?r rov 
ixoiiivav, ovrag uxuvru ru f^yovii^vu XQog Smvta 
r« i%6{uvu' ioriv uqu ag ro AH XQog ro AK, 

20 oikag ro AB xgog ro AE, icov de ro liiv AH 
r^ r, ro dl AK r^ Z' iortv uqu ag ro F XQog 
ro Z ovr (og ro AB ngbg ro AE, 

Tu UQU iiigri rotg douvrag noXXtackaoioLg rbv 
uvrov i%H Xoyov Xritp^ivru xuruXkrika' oxbq idet 

26 dttiui,. 

XV: FappuB V p. 338, 4. 

tf. l0xh\ m. 2 F. 7. htCv F. 8. nsyiJ^si V. 11. 

*^tf tu tm Z] in raa. m. 2 V. Z] seq. ras. 3 litt. V; Z 

lityiiyfi Jip. 12. OB] ©E9 (non F), BG B. 13. KA] 

HA \. tau uXXtiXoi9 V. iatlv B. 14. aUijXofg] om. V. 



ELEMENTORUM LIBER V. 45 



XV. 



Paxtes et similiter multiplices eandem rationem 
habent suo ordine sumptae. 

Sit enim AB magnitudinis F ei ^E magnitudinis 
Z aeque multiplex. dico, esse V: Z — AB : jdE. 

nam quoniam AB magnitudinis F et ^dE magni- 

jT Q tudinis Z aeque multiplex est; 

-^" — ' — > — 'B Ti — ^i quot sunt in ^J5 magnitudines 

. ^ ^ ^ „ magnitudini F aequales^ tot 

etiam in ^dE sunt magnitudini 

Z aequales. diuidatur AB in partes magnitudini V 

aequ^es^ AH^ H@^ ®B^ei jdE in partes magnitudini 

Z aequales^ JK^ KAy AE, erit igitur numerus mag- 

nitudinum AH, H&, @B numero magnitudinum ^K, 

KA, AE aequalis. et quoi^iam AH — H@ = @B 

et AK = KA = AE, erit AH : AK = H@ : KA 

= @B : AE [prop. VII]. quare etiam ut una prae- 

cedentium ad unam sequentium; ita omnes praece- 

dentes ad omnes sequentes [prop. XII]. itaque AH 

: AK = AB : AE. uerum AH = T, ^K = Z. 

itaque 

r:Z = AB:AE. 

Ergo partes et similiter multiplices eandem ratio- 
nem habent suo ordine sumptae; quod erat demon- 
strandum. 



iaztv B. Sh xal ra] dij seq. lacuna q>. 16. ®JB] BS F. 

yiE] post ras. 2 litt. P. 21. to] corr. ex tcS m. 1 p. 

JK]j in ras. m. 2 P. Z] corr. ex JC m. 2 I'. 24. 

Usi BFVp. 



Eav TteSaQa [isyi&tj avai.oyov _^, 

a^ avaloyov iszai. 

"Eera tieeapa (ityi&ri avaXoyov rii ^, B, r, . 

h dis zb j4 HQog ro B, ovTcag tb F wpos ro z/' idy^ 

OTt xal ivalka^ {avukoyov] iezai, mg to A xpb$ i 

r, otiTws tb B TCQos ro ^. 

Eil-^f&cj yap Tav fifv A, B leaxts aolXttJCiLaaS 
Ta E, Z, tmv 6h F, A aXXa, d fruj;£v, ledxig i 
10 nXdeia t« H, ®. 

Kal inil ledxts ietl TtoKXuxXdeiov tb E rot 
xal ro Z rov B, ra di fif'()i] rotg maavta>s xollantXa- 
ff^tg row awTov ix^i Xoyov, Sotiv &qk mg ro A irpog 
ro B, ovTag to E xqos tb Z. as ^s eo A spog ro 
16 B, ovTtas ro F ^rpog ro A' xal ag dga ro T Kpog 
ro ^, oiJrtos ro £ npoj ro Z. MrtAtv, ifff! ro H", 8 
rtow r, ^ iedxis ietl aoXXaitkdeiu, ietiv npa eSs to 
r ;rpos ro iJ, ovTOjg ro if wpog r< &. dtq d\ ro f 
jrpog To A, [ouriag] to E «pog ro Z* xai as aQ» zb 
20 £ jrpog ro Z, oijrcas ro H itgbg ro 0. ^av 3l riff- 
Uttpa (iByi&i] ttvdXoyov jj, ro di jrptoToi' tov TpArou 
[iBttov r), xal ro SEvtSQov rou TBtdQTov jist^ov ierat, 
xav ieov, teov, xuv iXttttov, iXartov. tl «pc ijiEQ- 
i%ii tb E Tov H, vntQixti. xal tb Z tow &, xaX el 
26 teov, teov, xal el iXattov, iXuTtov. xaC iatt t« ^%v 
E, Z TMV A, B ieaxiq noXXaxkdeitt, za Sl H, ® tav 



'. SeraL] hxtv P. lo'] (alL) C 
9. a] anpra F. 11. |arq 



S. omloyov] om. _. j _. __, , — , 

8. yafl] gnpta F. 9. n]^anpra F. ^ 11. iaiQc — _(, 

xollajilttoioy] -ov in ras. P. 13. layov] P; idyov "^IV" 

»ivTa KazdHijla Theoa (BFTp). 15. oSTms] eupra p; om. B. 

16, Z] corr. ex S m. 3 V, H, 0] 0, H Bp. 17. noUtt- 



ELEMENTORUM LIBER V. 47 

XVI. 

li quattuor magiiitudiiies proportionales sunt, etiam 
I permutando proportionalea erunt, 

Sint quattuor magnitudines proportiouales A, B, F, 

Pi ^ ^, ita ut sit A : B =^ r : ^. 

. dico, etiam permutando esse 

«1— 1_( sumantur enim magnitudi' 

num A, B aeque mnltiplices 
E, Z, magnitudiuuiu autem r, d aliae quaeuis aeqne 
multiplices H, ®. 

et quoQiam £ magnitudinis A t\> T^ magnitudiniB 
I B aeque multiples eat, partes autem et aimiliter mul- 
iaplices eandem rationem habent suo ordine sumptae 
[prop. XV], erit A:E = E\Z. uerum A:B = T: A. 
qnare etiam r : A = E : Z [prop. XI]. ruraus 
quoniam H, & magnitudinum F, ^ aeque multi- 



kplices sunt, erit F: A = H:® [prop. XV]. uerum J 

F: A •= E:Z. itaque etiam £ : Z = H: [prop. XI]. I 

si autem quattuor magnitudines proportionales sunt, 1 

I 

A 



L si autem quattuor magnitudines proportionales sunt, 
et prima maior est tertia, etiam secunda maior erifc 
quarta, et si aequalis, aequalis est, et si minor, minor 
[prop.XrV]. itaque siBmagnitudinemi/superat, etiam 
Z magnitudinem @ superat, et at aequalis, aequalis est, 
et si minor, minor. et E, Z magnitudinum A, B 

wlKaitt] seq. ta S\ fiioti toij Awvxias xoiXciJtXaeiois tov av- 
xav Ixti loyov Ijiifd^httt ^axuXXjila Bp. 18. F] in ras. 

m. 1 p. (DE *e] alX' a>t F, 19. ovt«s] om. P. 20. to] 
(alt.) e corr, V. 23. iXaoaov, iiaiKsov V, 34. ©] aeq. raa. 
1 litt. V. «al ffl %av Theon (BFVp). 26. Nal tl] *av 
Theon (BFVp). imv P. 26. xa Bf — p. 48, 1: itoXXa- 

)[la«ic(] mg. m. rec. p. 



48 



ETOISEIiiN I 



f, id oAAa, « Ijvxtv, {attxig itolKaalaaicf lativ agt 
mg ro A repdg ro F, ovrrag ro B Jtgbg ro z*. 

"fiftv apa rieOKQa iisyi&rj acaAoyoi' ^, xal ^rffJ 
Aa| avtt?.oyov lazar owep ^'6£( 5£r|tt(. 

'Ekv avyxsifitva fiayid^ti avaioyov t/, 
diaiQi&ivra avaloyov latai. 

"Eatcj evyxBifiEva (i8yi&ij avaloyov ta AB, BB 
Vjd, ^Z, ois TO AB Jtpog To B£, otfTcis to VA «po<| 
10 ro li/Z" kiya, ort xa! diaiga&ivta avaXoyov iexai 
atq to ^S ffpog 10 EB, outras ''^o PZ srpog ro <^ZH 

^/AiJqD*» j-Kp tmv iihv AE, EB, FZ, ZJ icaxts 

aoXXajtlaaia ta H@, &K, AM, MN, Tt5i/ Sb EB, 

ZA alltt, tt irv^Bv, lattmg xolXaakiiaia ta KIS, NII. 

3 Kal inel iaay.iq iarX xoXkttxXaaiov t6 H& rou 

AE %ttl ta ®K Tou EB, (oaxig aqa iatl itolXastXa- 

oiov To H® Toij AE xal ro HK tov A&. {aaxL$ 

6i iart aoXXanXaat-ov to H& tov AE xal t6 AM 

toi) rz- lattxig aga iatl TrolXaTcXaaiov t6 HK tov 

20 AB xal rb AM tov TZ. xdXiv, ixsl laoaus iotl 

itoXXaitXdaiov t6 AM tow FZ xal t6 MA'' tou Z^, 

ladxts liQa iatl noXXoTcXdaiov ro AM rov FZ xal 

ro AN roii rz/. Cadxis Se ^v reoiiareAaBiOV t6 yiM 

Tow rz xttl ro ffff low AB' ladxig aQa itStX sioA- 

B AaffAa*T(of To ifA" ror AB xa\ to AN rov VA. tk_ 

HK, AN ttQa twv AB, EA ladxig iarl «oXlaaXdeH 

1. al aupra m. 2 F, 11. £B] BE Bp, et 'V e 

t6 dZ] TO ZJ F, V m. 3} JZ P. 12. EB] snpra. 

S F. 17. H.K] H in rag. m. 1 V. AB] A e corr. m. 2 

18. AM] in ras. m. 2 T. 19. FZ] F m ras. m. 2 



^sa^ 




^m ELEMENTORUM LIBEB V. 49 

aeque multiplices suut, et H, @ magnitudiiium F, /i 
aliae quaeuis aeque multiplices; itaque A -.r = B\ /1. 
Ergo si quattuor maguitudiues proportiouales sunt, 
etiam permutando proportionalea eruat; quod erat de- 
mouBtraudum. 

XVII. 

»Si compositae magnitudines proportionalea sunt, 
etiam dirimendo proportionalea erunt. 
Sint compositae magnitudines proportionales AB, 
BE, rj, JZ, ita nt sit ABiBE = r^: JZ. dico, 
etiam dirimendo ease AE : EB = FZ : AZ. 

sumantur enim magnitudinum AE, EB, FZ, Z^ 
aeqne multiplices H&, ®K, AM, MN et uiagnitudi- 
num EB, ZA aliae quaeuis aeque multiplices KIS, NIT. 
et quoniam H® magnitudinis AE et &K magnitudi- 
nis EB aeque multiplex est, erit H® magnitudinis 
AE et HK magnitudinis AB aeque multiplex [prop. I]. 
nemm H® magiiitudinis AE et AM magnitudinis VZ 
aeque multiplex eat. itaque HK maguitudinis AB et 
AM magnitudinia FZ aeque multiples est. ruraus 
quoniam AM magnitudinia FZ et MN magnitudinis 
ZA aeque multiples est, erit AM magnitiidinis rz 
et AN magnitudinis FA aeque multiplex [prop. Ij. 
erat autem AM magnitudinis FZ et HK magnitudi- 
nifl AB aeque multiplex. itaque HK magnitudinis 

IAB et AN magnitudinia FA aeque multiplex est. 
:jS(o] in ras. m. 2 V. HK\ K i 

30. AB] B in ra» m. 2 V; TZ P. 
.^B P. TcdUv iitii - 21: Tov rZ] 
21. ZJ] JZ BVp. 23. AN] AH V 
jtoi] (prius) Iiie p. AS] eraa. p. 
1 Bnolldoi.sdd, IlBiberafl MBnge. II. 





50 ETOIXEIfiN t'. 

xaXiv, insl lOaxiS iotl iroXXaniaaiov ro @K 
xal ro MN tov Z^y, ffftt di xal ro KS Toi) EB 
iodxts mlXanXdoiov xal ro NII roii Z^, xaii Ovv- 
TB&iv to ©S roTJ EB isdxtg iott noii.a3t!i,ttaMv xal 
5 ro MII zov 7.£i. Kal ditti ioiLv ros ro AB itQog ro 
BE, ovrwg zo Pzl JTpos ro dZ, xal iCArjxtai rtov (tiv 
AB, r^ ledxig noi.Xani.dota za HK, AN, rdn di 
EB, Z^ iOttxis TtoXXajtXdoia xa &S, MIl, tl afftt 
vjtBqixtt tii HK tov &S, jmEQi%£t, xal xo AN roii, 

10 Mn, xal si Ceov, toov, xal ei ^Jlttrrov, iXartov. vttt 
tyixoi 8ii ro HK tov &S, xal xotvov dqiaigt&i; 
tov &K vitipixti affa xaX ro H& xov KS- dk' 
ti vntpetx^ 10 HK xov 0S, vasptCx^ xal xo AN ti 
MJI' vnB(fi%Et aga xal ro AN roiJ M/Jf, xal 

16 dtpmQeO-ivtos rov MN vnspixei »al x6 AM tov NIT- 
mtnt si vntffixst xo H& ro»; KS, vnEpixtt xal to jIM 
zov Nn. ofioiais iJij Sfi^oiitv, ort xav Coov 5 ro H8 
rro KS, ioov ioxai xal xo AM x^ NH, xdv HaTzov^ 
ilazTov. xai ioxt td fiiv H®, AM xwv AE, FZ 

20 iottxis noXXanXdeta, td ds KS, NH tav EB, Z^ 
aXXa, d txvxiv, iedxis noXXanXdoia' ioziv aga ajs 
AE 3tpos ro EB, ovtess ro TZ nQos to ZA. 

'Eav apa Ovyxtifitva (iiyi&ri dvdXoyov y, xal di 
(ft&ivza dvdhryov totaf ont(f tSti Stll^ai. 



1. iuTiv FT. 3. ZJ] ZB P. 4. to] Sfu 
add. m. 2 F. e, AZ] ZJ BVp. 7. AN] e corr. m. 2 

8. ZJ] ^Z P. Seq. in Bp: aUa u fivjtvi idem T m. 8, 
et F in raa. m. 2 (om S), aed omisHO ladms (fuit in mR. m. 
2, aed enan.). 10. llaaaov, Uooaov p. 12. oUa] alX PV. 

13. intpttiel PVp; imptixfv B; vnigfxct e corr. F. to 
HK tov fiS vKEqiixf] mg. m. 1 P. vntQc^it] p; ixtdtfxtv 
PB; ineeczit FV. AN] AH in ras. m. 1 p. 16. vntfixn] 
-ixct in raa. P. K3~\ in ns. V. 18. f<tia(] om. F. 



] 



■ ELEMENTOHUM LIBZR V, 51 

itaque HK, AN magnitudinum AB, V^ aeque multi- 
plicea sunt. ruraue quoniam &K magnitudinis EB et 
MN magnitudiiiis Z^ aeque multiples est, et KS 
magoitudinis ES aeque multiplex est ac NTl magni- 
tudinis Z^, etiam componendo &S magnitudinia EB 
aeque multiplex est ac MJl magnitudinia ZJ [prop. II]. 
et quoniam est AB -BE= fzf : ^JZ, et sumptae sunt 
magnitudinum AB, FJ aeque multiplices HK, AN, 
et magnitudinum EB, Z^ aeque multiplicea &S, MTI, 
ai HK magnitudinem ©S superat, etiam AN ma^ni- 
tudinem MII superat, et si aequalis, aequalis est, et 
si minor, minor [def. 5]. itaque HK magnitudinem 
®!S superet, et ablata, quae eoinmunis est, 0/f, 
etiam H® magmtudinem K^ superat. uerum s\ HK 
magnitudinem ®S superabat, etiam AN magnitudinem 
MII BUperabat [lin. 8 sq.]. ergo etiam AN magni- 
tudinem MH superat, et ablata, quae eommunis 
eat, MN, etiam AM magnitudinem JV/7 superat. 
quore si H® magnitudinem KS auperat, etiam AM 
magnitudinem NII auperat. similiter demonatrabimus, 
si H® = KS, esae etiam AM= NH, et ai H® < KS, 
.esse etiam AM <. NH. et H@, AM magnitudinum 
AE,rZ aeque multiplicea sunt, KS, IV/7autem magni- 
tudinnm EB^ Z z/ aliae quaeuis aeque multipHces. itaque 
AE-.EB^^rZ: ZA [def. 5]. 
Ergo ai compositae magnitudinea proportionales 
sunt, etiam dirimendo proportionales erunt; quod erat 
demonBtrandum. 

tlaeaov, IXaeaov Bp. 19. AE, rZ] rz, AE Bp et P erasi) 
r. 20. KS] KZ <f. 21. u] Bupra m. 2 F. 22. Z^J] Z 
in TM. Vj JZ Bp. 23. 5] iazai. V, snpra aox, m. 2 j. 



J 




tog 

i 



ETOISEIilN * 



'Eav dif]pr}(iBVce fitys^fi i 
avvtf&iVTtt avukoyov Sstai. 

"Earoa SiriQTj^iva fityc^i^ oivaXoyov ta AE, EB, 

6 rz, Z^, mg ro AE jrpog to EB, otJttas t6 FZ jiqos 

t6 Zi^' liya, on xal awre&fvTa avaloyov iorat, 

mg ro AB jrpoj t6 5£, ovrto? ro Tz/ Jipo? t6 Z^. 

JE^ yap fi»; ^ffrtj' rag t6 ^B wpog t6 BE, ovTtog 

t6 r*^ jrpog t6 ^Z, fffrat roj t6 AB stpog t6 B 

10 oCtios t6 r*^ ijTot repog iXaaaov xi rov /SZ ^ jrpi 

fiErgov. 

"Eara nQorsgov «pog {'Aaeooi' t6 z/if. xai iwf^ 
tariv rog t6 ^S 3tp6s t6 BE, ovTag ro /'z/ ffp6s t6 
AH, avyxELfieva (lEyt&rj avaXoyov iartv' wOrt xat 
15 Hiaiqh&ivra avakoyov ierai. EGriv a^a ros ro AE 
ngog t6 EB, outoj to m irpog ro H^i. vttoxBixai 
fil Kal mg t6 j4E npos t6 EB, oures t6 /"2 Bp6s 
t6 Zzl. xal ag apa t6 FH Ttgog t6 ff^, oijtojs t6 
rz ffpos t6 Z^, fiei:tov de t6 jrpratuf t6 FH toi- 
20 rgCzov tou i^Z' f^tft^oi' «pa xai t6 Sivteqov t6 if^ 
ToiJ T£T((pTOu Tou Z.^. «AAa jtal ^iaTTOV onEp ^ffr!i' 
(JdiifaTOJ'" owx «pa ^ffrij' (as t6 j4B TtQog t6 B£, 
ouTws t6 r^ TtQog ilttOCov Tov Z^, ofioiag Sij Ssi-_. 
^oiiBV, OTi ovSi nrpos (ibi^ov 3rp6s «vro Spu, 

4. ^El -* PBJ!'V. 5. rZ] (priuB) r PBPV. 6. ZJ] 
JZ F. 7. id] (alt.) oiD. P. ZJ] JZ P. 9. td] (alt.) om. P. 
^Z] PF, V m. 2; ZJ Bp, Vm. I. tos to — 10: to FJ] 
mg. m. 2 V. 10. ^kaeeov n] ilaTTov ip, eupra sor. ti m. 3. 

ioij]to zov F. JZ]PF, Vm. 2; Z^Bp, 12. nartov P. 

13, lis "1 010* p, nt iam lin. 9 et postea saepins, 
liQ if. T<i] (qnartnm) om, B. 14. ^orit'] e c 



i 




ELEMENTORUM LIBER V. 



53 



Ir 



XVTII. 

Si diremptae magmtudines proportionales sunt, 
L compositae proportionales erunt. 

- Sint diremptae maguitudi- 

H^— iB nea proportionalea AE,EB, 

Z g ^^ rz, ZA, ita ut sit ^£; : ER 
= rZ : Z^. dieo, etiam 
impositas proportionales esse, 

AB:BE=r^:ZJ. 
nam si non est AB:BE= VJ : z/Z, erit ut AB 
BE, ita r^J aut ad minus magnitudine JZ aut 
madns, 

priaB ad minus ^if aequalem r^tionem habeat. et 
t[uoniam eat ^45 : BE = VJ : AH, compositae magni- 
tudines proportionalea sunt. quare etiam diremptae 
proportionales erunt [prop. XVII]. erit igitur 
AEiEB = TH: HJ. 
ippoeuimus autem, ease etiam AE : EB = FZ : Zjd. 
[nare etiam m : HJ = FZ ; ZJ [prop. XI]. sed 
prima rH. maior est tertia TZ; itaque etiam secunda 
HJ maior esfc quarta ZJ [prop. XIV]. uerum etiam 
minor est; quod fieri non potest. itaque non est 
ut AB ad BE, ita fz/ ad minus magnitudine ZA. 
similiter demonstrabimue, ne ad maius quidem aequa- 
lem rationem habere T^. itaque FJ :ZJ = AB : BE. 



rH] FB q, (non F). 18. ZJ] dZ F. 
H ZJ] mg. m. a V. 18. w] (tert.) om. 

4. 2, led corr. 21. iCTapiov] in rae, p. 

(^«ticw F. Z4I m rsB. in. 2 V; ,dZ E 






-10: 



m 



'Eav apa diiiQijii^va (uyi&^ avaKoyov jj, xi 
xt&ivxa avaloyov iszaf o««p iSei Set^ai. 

'Eav 7) rag oAof jrpoff oAor, ovras atpatife&h 

5 Kpog a^atQE&iv, xal tro Aoiirot' «pog t6 Aot^ 

jtox' lerai d>g olov jrpoj oAov. 

fiBro) ycp 0)5 o/of to j^B itpoff o/ov to I^i^j 
rws ayatpffl-fi/ t6 j4E repog a^iafpEO^v ro fZ* Ji^ye 
ori xa! Aofjrov t6 EB aQog konthv t6 Z^ ^iTTat AA 
10 oAov To AB srpos oAok t6 r^. 

'EjtEl yaQ ioziv dtq z6 AS spog t6 V/], ovttoq » 
AE jrpo; t6 fZ. xa\ ivalXa^ s>q tb BA ngbs 1 
AE, ovzeys t6 AF repog ro TZ. xa! iAsl Ovyxfifitvafl 
(isyid^ avakoyov iortv, xal SiaiQBQivra avdi.oyov' 
15 IgTtti., mg t6 B£ jrpog to EA, ovtag t6 .JZ irpog 
ro rZ' xttl ivaXld^, <ns r6 BE wp6g t6 ^Z, ovrmg 
t6 EA JTpog to Zr. Mff d^ ro -^B irpos t6 FZ, oiVs 
rojff vxoxsittti oKov zo AB wpog o^Aov to FA. 
J.0MOV apa t6 EB irpog Aoinof r6 Z^ iUTo:! ms okoi 
sn t6 AB jrpog oAok t6 F.4. 

Ettv uQa jj tag oXov n^pog oAov, ovtas a^tttQs9ii 
n:p6s d<pKips9iv, xal tb lomov jrpog t6 AoiJtoi 
as oAov n:p6s oAov [o:jr£p i^tfet tfer^at]. 

[ffai ixsl iSeCx&fi a>s t6 .^B srpoff t6 F^, ovrta 

BQ To EB XQog t6 Zz?, «al ivaKka% tag tb AB «pog i 

B£ owTOjg t6 rj ffpos t6 Z.^, awyxetfievo; apa (isyi&i 

dvdloyov ioriv iStix^^ Sh ats i^o BA wp6s to AS 

oiVoJS t6 ^r Jrpoff t6 FZ' xai iaziv dvaffzQsil-avria 



9iv 



]j] iatat tp (non F). 2 
j .1E tii/osi mg. m. 2 F. 



tttt] eraa. F, 



ELEirENTORUM LIBEE V, 



55 



Krgo ai direniptae magnituilinea proportionales 
snnt, etiam compositae proportionalea eruut; quod erat 
demoDstrandum. 

XIS. 

Si totum ad totum eandem rationem habet atque 
iltblatnm ad ablatum, etiam reliquum ad reliquum ean- 
^em rationem habebit ac totum ad totum. 

8it enim AB: Fd = AEi FZ. dico, esae etiam 



EB:ZJ = AB-.r^. 

nam quouiam eat AB : Fz/ = AE : VZ, etiam per- 
mutando est BA: AE '^ ^T: TZ [prop. XVT]. et 
KC|QOtiiam compositae magnitudiues proportionales suut, 
Rtotiam diremptae proportioualea erunt, 

BE:EA = zJZ: FZ [prop. XVII]. 
et permutando [prop. XVI] B£ : .irfZ = E^rZT. aed 
aupposuimus, ease AE : FZ = AB : I^irf, itaque etiam 
EB:Z^ = AB:rj. 

Ergo si totum ad totum eandem rationem babet 
atque ablatum ad ablatum, etiam reliquum ad reli- 
quum eaiidem rationem habebit ac totum ad totum; 
^uod erat demonstrandum. 

telo»] (alt.) m. 2 V. 11. iaii <p (non F), olof ro AB neoe 
"*■ TO Theon (BVu, F euan.). la. J F} PJ P. 14. 

I Fi ^9« PBVp. 16, Poat (jc add. apa Pm. rec, 

2; Bp. 16. rZ] Zr P. ivalla^ aga laciv Theoii 
FVp). 19, ZJ] JZ P. 21, ngos difttifMv] mg, F. 
24. ffoeffif mg, m. 2 V, «ul fjiil] euan., del, m. 2 F. 
"- i ZJ] Z.J P. 26. 10 ZJ] F; ZJ P; To JZ V 

ras. 27. iativ] in raa, m, 2 V; larai Bp, iJi xo! 
10 AE] AE Bp. 28. ro TZj FZ Pp, 



56 



ETOIXEIiiN f 






tv 

M 



noQiefitt. 

'Ex S^ tovTov ipttvfpov, oTt ittv evyxsifiEVtt 
avttioyov jj, xal KvaeTptii-avri KvaXoyi 
iSsi SeI^ki. 
B ^. 

Ettv y T^Ctt fiEys&ij xkI nAAa avzots (ea za 
nk^^os, GvvSvo Xttfi^avoiiEVtt xttl iv xip avx^ 
Xoya, SC Eeov Se to Jiptoroj' tou tp^Toi» ^Bt%ov 

Tj, Xal ro xitttQTOV TOti fXtOV ^Et%QV lati 

10 xav tOov, fffov, xav ilarrov, Iktttrov. 

"Eerm ipitt iityi&ij ro; ^, B, P, xul alXtt avtol 
iea ro nlij&og ra z/, E, Z, evvSvo Kafi^avofiEVtt iv 
T^ avrp Xoyp, rag [ilv to A XQog to B, avttag t6 ^ 
;rp6s t6 B, ms S^ ro B Ttgos t6 F, oifTtos ro B »967 

16 ro Z, tf(' ftfou 5i fiet^ov iexia to A %oi> P- Xiymy oti 
xttl To ^ tov Z ftffgov lettti, XKV l'6ov, teov, xav 
IXttTrov, SXaTrov. 

'EiieI yap fiEtt^ov iari t6 A roti F, liXXo Si ti to 
B, t6 Se [LEt^ov xpog t6 kuto fiei^ova Xoyov %*t 

20 JjJiep ro iAarroi', t6 ^ «pa repoff t6 B (lEi^ovtt Xoyov 
i%Et ^KEQ t6 r" «pog 16 B. ttXX' ms ^*v ro A jtQos 
t6 B, [owrros] t6 A ngos t6 E, tos Se t6 r" Jtpog ro B, 
ttvdnaXtv oiirojs ro Z npog t6 £■ xal ro ^ apoc npof 
t6 E pL.Ei%ova Xoyov ix^t ^aep r6 Z npog ro E. reav 



1. ndeifffia] tng. PFBp; om V. i. Seq. acholiams n, 
app. 7. »«f'] Offl. p; m, 3 B. 10. aa»] xa! ^o* P. %Sp] 
laiai, Nccl Juf P. naaeof, ^Xaaaov Bp. 12. xnl iv Bp; 
na/ Bupra m. 2 F. 14, E] (alt) ante ras. 1 htt. V. 17. 

IXaaeov llaeaav Yp, 31. dlla B. 33, ovitae] om. P. 

to E] E P. xo r] r P; to add. m, rec; m Z 91. xi 
B} S F; T» E q). 23. avaTiaXiv] «al zo A 7. to £] £ qr; 
Bequentia euan. F. 



ELEMENTORUM LIBER V. 



57 



Corollatium.') 
Hiuc maoifestum est, si compositae mf^itudines 
Fproportioiiales sint, etiam conuerteado proportionales 
f eas fore. — qiiod erat demonstrandum. 

XX. 

Si datae sunt tres magDitudines et aliae iis numero 
Wquales, binae aimul coniunetae et in eadem propor- 
tione, ex aequo autem prima tertia maior est, etiam 
quarta sexta maior erit, et si aequalis, aequalia erit, 
et si minor, minor, 

Sint tres magnitudines ji, B, F et aliae iia numero 

_^__ aequalea ^, E, Z, binae coniunc- 

tae in eadem proportione, sci- 

'_^ ^ ^^ licet A : B = J : E, et B-.r 

' ' = £ : Z, et ait ^ > F. dico, 
se etiam z/ > Z, et si .^ = J", esse .i;/ = Z, et si 
( < r, esse J <Z. 

nam quoniam A^ F, et alia quaeuis magnitado 
Ht B, et maius ad idem maiorem rationem habet 
^uam minuB [prop. VIII], erit A : B'^ T: B. uerum 
: B = ^ : E ei e contrario [prop, VII coroll.] 
r.B = Z:E. 

1) QnEie praecedimt uerba p. 65, 24 — S8 immerito ab Sim- 
BODO aliiBque nituperantarj nam uersim contiiient demonstra.- 
tionem comierBae rationiB. (lemoostrauimuB enim (p. 56, 19) 
AS : rj =- EB : ZJ, nnde AB : EB ^ TJ : ZJ; aed simul 
erat (p. 65, 12) BA: AE = JT : TZ; tum u. def. le. niMlo 
minus bic locus interpolatus eaee uideri potest (eed ante 
Theonem), quia EnolideB aumquam corollarii rationem reddit, 
id qnod ipeiQB uoeabuli no^mfia notioni (Froclus in Eucl. 
p. 301, 303) B,duereatur, huic loco similie eet iuterpolatio 
TheoniH poat V, 4. 



E^gova AOjJ^^^ 



58 ETOIXEIflN (' 

Si ffpos tb «VTO loyov ixovTcov to fiei^oi 
ixov fitt^ov iOTiv. (lettfiv uqa xo ^ rou Z. ofioimi 
Sil dti^oiitv, oTi xav fffov ?; to j4 roi r*, rffoJ' ^ffTai 
xttl ro ^ rp Z, xav cXcertov, SlatTOV. 
5 '£«!' Squ 37 rpi^a (iByi&rj xal aXXa avrots ftfa ro 
jtA^dog, evvSvo XafL^avoyLtva nal iv ra avra Xoyt^^ 
ii taov Se xh nQiarov tou tqCtov (ieI^ov y, xal i 
tita^TOV ToO txTov (i£l%ov iatai, xav iaov, tOov, 
iXattov, iXtttzoV oneQ iSsi Ssi^ai. 



10 xa . 

'Ettv Tj Tp^ft fiByiQ->i xal aXXa avtots toa tb 
nX^&os avvSvo Xa(i^av6(i.Bva xal iv rro avt^ 
Xoyra, tj Ss TBtapayitivtj avtav t] avaXoyia, 
St laov Sb to it(fatov TotJ tqCtov ftEtgoi/ 
h To titagtov roii sxrou ^et^ov iatai, xav fffo' 
teov, xav eXattov, iXattov. 

"EaTco tqCu HEyiQti ra ji, B, P xal aXXa avTots 
laa To nA^d^og T« ii/, E, Z, evvSvo lafi^avofiEva xal 
iv ta avrcS koya, iatm 31 tftagaynivti avtmv ^ 
20 avakoyCoc, ms fiiv to A nQog t6 B, ovtms to E npos 
t6 Z, ats Sh to B npog ro r, ovrojg ro -4 wpos ro 
S, iJt taov Sh To v^ roii r fiei^ov iatm- Xiyco, oti xaZ 
ji Tou Z f(£[£ov laTUL, xav taov, taov, xav iXi 



i 



TOV, fiOTrov. 



.««I^p 



iftim Theon (BFVpl. ' ^on»] ] 
_fi£fEo»'] cort. es p^i-E»* V. 3. t6 



ioaffof p, 8. loBV fotea, 

P. 9, flaaaov, tiaaaov p. 16. rtuoflo», llaaaov FVp. 
1. atyi^Tj dvaloyav PBFVpi con-. Gregoriua. to] e 

■. V m. 2. 19. b] om. B; euan. F: toe w. 22. ro ..*] 



ELEMENTOEUM LIBEH V. 



59 



XSI. 



itaque etiam ^:E>Z:E. eorum autem, quae ad 
idem rationem habent, maiua est, quod maiorem ra- 
tionem habet [prop. X]. itaque z/ > Z. aimiliter de- 
monstrabimus, si A = T, esae etiam A = 1, et si 
A <,V, esse etiam ^i < Z. 

Ergo ai datae suat tres magnitudineB et aliae iis 
numero aequales, binae simnl coniunctae et in eadem 
proportione, ex aequo autem prima tertia maior est, 

tetiam quarta sexta maior erit, et si aequalis, aequalis 
«rit, et si minor, minor; quod erat demonstraudam. 
* Si datae aunt tres magmtudiues et aliae iis numero 
aequales, binae simul coniunctae et in eadem pro- 
portione, et perturbata est earum proportio, et ex 
aequo prima tertia maior eat, etiam quarta sexta 
maior erit, et si aequalis, aequalis erit, et si minor, 
inor. 
gint tres magnitudines A, B, I' et aliae iis uu- 

mero aequalea A, E, Z, 

binae simul coniunctae et 
in eadem proportione, et 
perturbata sit earum pro- 
ita ut sit A : B = E : Z et B:r= A: E 
[de£ 18], et ex aequo sit A > F. dico, esse etiam 
"^ > Z, et si -< = r, esse ^ ■= Z, et ai ^ < T, 
esse A <. Z. 





60 



ZTOIXEIKN i 




'ExiX ytt(f nit^ov ievt ro A tov F, cAAo di xt xi 
R, to A aga Kpog ro B [leilovcc ^oyov Ix^t ijneif to 
r Jtf/og ro B. «/lA' roj itiv tc A Ttpos id B, ovrcag 
ro E srpos t6 Z, cSg tfe to f «pos' ro B, dvaitaXiv 

5 ourrag ro E stpos to z?. xal to £ «p« irpos ro Z 
[iBt^ova ioyov f^*' ^Wp ro £ jrjios ro ^. 3i()6; o 
if ro avro ftciflova idyov Ijjst, ixtivo ilaeeov lativ 
iiaOOov a^a deil ro Z tov .^' fi^E^ov npo: ^flrt ro ^ 
rot' Z. oftotfog d^ dftlo^ff, ort xnv ruov ;/ t6 ^ t^ 

f, /'tfov (ffrat xal to ^ t<n Z, xav iXattov, iJiMtTt 

'Eav UQa y rgia fiefi&t) xal &KXa avxolq fffa 

Ttkri&osi evvSvo Xaii^avoiisva xal iv rp avx^ ^^yipi 

^ Si titaffaynivtj avrav ^ dvaloyia, 6i i'6ov Si to 

irptorof Toij tQiiov (lEi^ov tj, xal ro TErupTOi' rot 

S EXTOV ftet^ov iUrai, xau teov, feoi", xav ikarrov, iXtt%^\ 
ToV oaBQ ^Stt Sttlai. 

'Eav t/ oxasaovv (iByi9ij xal alXa avron 
HOtt ro xi^Q^og, dvvSvo kafi^avofieva xal 
ta ttvra \6y^, xal St HSov iv ra avtp i^oyA 



r^ 



"Eara oTtoeaovv iisyi&rj ra J, B, F xal uXXa 
totg fff« t6 hA^&os r« jJ, E, Z, GvvSvo Xaftpavoftevte 
iv Tco a^rto ^oyip, mg {U^i' to A ngog rd B, ovttag 
5 ro /f «pog ro E, thg Si t6 B wpog t6 F, oStcus ro E 
TtQoe to Z- Xiyta, ort xal Si foou iv ta avta Aoyji 
iatai. 

] seq. raa. 1 litt, V. 7. ^Kiifo] -o 
F. 8. Xlaatsov'] om. F; tlanov B. 
t. 5] om. B. 10. xa/] om. F. ft«B- 
I. 'b'] om. 9. ««/] i nai FV. 



2. -i] Bupra P. 



4 



ELEMENTORUM LIBEH V. 



rnam quouiam A > F, et alia quaedam magQitudo 
t 5, erit ^ : B > r : a [prop. VIII]. uerum 
J:B = E:Z. 
et e contrario [prop. VII coroll.] F: B = E: ^, ita- 
que etiam £ : Z > E : .c/. sed ad quod idem maio- 
rem ratioueni haljet, id minus eat [prop. X]. itaque 
Z < ^. quare i/>Z, similiter demonatrabimus, si 
A = r, esse etiam ^ = Z, et si ^ < J^, eaae ^<.Zi. 
Ergo si datae suut tres magnitudiues et aliae iis 
numero aequales, binae simul coniunctae et iu eadem 
proportione, et perturbata est earum proportio, et ex 

Iaequo prima tertia maior eat, etiam quarta aeicta 
(Oaior erit, et si aequalis, aequalis orit, et si minor, 
JBainor; quod erat demonstrandum. 
XXII. 
* Si datae sunt quotlibet magnitudines et aliae iis 
ilumero aequales, biuae simul coniunctao et in eadem 
proportione, etiam es aequo in eadem proportione erunt. 
Sint quotlibet magnitudines A, B, V et aliae iis nu- 



e, , — I ^i-i-i—i jvi 1 < 

mero aequales .d, E, Z, binae simul coniunctae in eadem 

proportione, ita ut ait .,4 : B = z/ : E et B: r= E:Z. 

vdico, eas etiam ex aequo in eadem proportione fore.') 

y 1) H. eT^ : r — ^ : Z (def. 17). 

15. iltusvov, nttaoov V. 19. %ai] om. Bp. 26. ro] 

fprinmm) -d in raB, m. 1 B. 27. /oovtat Bp, Dein add. 
Theoa: ms to A jtsos to T, ovri»e ro J jrpos ro Z (BFVp; 
om. P), 




62 STOIXEIHN i'. 

EH^ip^ci yicff Tav (liv A, /t ieaxtg jioAAi 
ra H, &, Tmv di B, E aila, a hvxev, iaaxis Jiolla- 
nXaaia ta K, A, xal Szi tmv F, Z «AAa, a Itvxsvj 
iedxts aoXXanlaeia ra M, N. 
B Kal inti ioxtv log lo A n^bg ro B, outMg ro 
TiQog ro E, xal tCX^arai %mv (liw A, /i leaxig roAJUkI 
nXaeia xu H, &, rdv dl B, E akka, a izviiv, Cedxtg 
7toKKax).d<sta za K, A, iOtiv apa tog to H ■KQog tii 
K, oTJTrag to @ Jtpos ro A. Sia ta avta d^ xai mg 

10 ro K jrpog to M, ovtag to A xpoq to N, iittl 
tpia (ityi&t) ietl t« H, K, M, xaX aXXa avxotg tt 
to nXri^og t« 0, A, N, evv8vo lafi^avontva xal iv 
xa aiita Xoyip, St t6ov a^a, ti viiepixti to H roij M, 
vnsgixti- xal to @ rotJ N, xal tl teov, (eov, xal ci 

16 lAfttrov, iXattov. xai iatt ra fiiv H, ® ttov A, A 
laaxtg xoXXaaXdeia, xa Si M, N ttov P, Z dXXa, 
ixvxtv, ledxig xoXXaxXddta. i^dtiv a^a mg xo A xgog 
tb r, ovTtas 10 ^ ffpog To Z. 

'Edv dga jj onoaaovv iteyi&ri xal dXXa avxotg fffa w 

20 ff^^S^os, evvSvo Xa(j,^ttv6it£va iv rra avtm Adycj, 
St i6ov iv tra avta Xoya ioxaf ontQ eSti Sstiat^, 



xy . 

'Edv ^ tpia (liyi&ti xal aXXa avtotg iaa t( 
xX^&og avvdvo Xaft^avo^Eva iv xa avx^ Xoyip. 
es 17 S^ TETafayfiivtj avtav 17 dvaXoyia, xal St' 
taav iv ta avTa Xoyet iaxai. 



UwM 
txtg 

M, 
cl 

a 

tog 



. Se} om. p. In P in hac pag. complara euan. 5] • 
''? 3. tt] om. F. 6. jipofi id] in raa. p. 7. Si] 

c, p. aj m. 2 F. 9. Wfloe] om. 9. 12. td 0, A, 
m. pi m. 2 V; mg. m. rec. B. 16. iXaeaov, iluaeov p. 



I 



■ ELEMENTORUM LIBER V. 63 

Sumantur enim magnitudinum A, jJ aeque multi- 
plices H, 6f, et maguitadiuum B, E aliae quaeuis aeque 
multiplices K, A et praeterea magnitudinum T, Z aliae 
quaeuis aeque multipliees M, N. et quoniam eat 
A : B = ^ : E, et sumptae sunt magnitudinum A, A 
aeque multiplices H, @ et magnitudinum R, E aliae 
quaeuis aeque multiplices K, A, erit H : K = &: A 
[prop. IV], eadem de causa etiam K:M = A:N. 
iam quoniam datae sunt tres magnitudiaes H, K, M 
et aliae iis numero aequales &, A, N, binae simul 
coniunctae et in eadem proportionej ex aequo, si H 
maguitudinem M superat, etiam magnitudinem JV" 
superat, et si aequalis, aequalis e8t,*et si minor, minor 
[prop, XX], et H, & magnitudinum A, jJ aeque mul- 
tiplicea sunt, M, N autem magnitudinum F, Z aliae 
quaeuis aeque multiplicea. itaque A : r= A : Z [def. 5]. 

Ergo si datae sunt quotlibet maguitudines et aliae 
iis numero aequales, binae simul coniunctae in eadem 
proportione, etiam ex aequo in eadem proportione 
erunt; quod erat demonstrandum. 



XXUI. 

S! datae sunt tres magnitudines et aliae iis nu- 
mero aequalea binae simul coniunctae in eadem pro- 
portione, et perturbata est earum proportio, etiam ex 
:#«quo in eadem proportione erunt. 



■ni6. «] m. 8 F. 18. r] in rag. iB. 2 P. J] in ru 

"tIP. Poflt Z in P add. nal it b««6_ (afla Uxiv mg. m. 1 

%h A nifoi zo d (in rae. m. 2), ovxeit td r (is ras. i 

n^hi xo Z. 23. i] om. p; m. 2 B. 24. Supra h 

¥.ai F. 26. iaovxui BFVp, 



64 ZTOISEIiiN s'. 

"EOTa tptte fiByi&t] tce A, B, P xa\ aXlce avrots 
Caa zb zlij&os avvSvo Xafi^avofitvtt iv ra avtp Xoyp 
zu ^, E, Z, eoza 61 reTopayfteVij avTiDV tj avaloyia, 
(ug fiiv tA j4 jr4»6s t6 B, ovtioe ro E «(tog t6 Z, eog 
5 (Jf t6 S Jtpog r6 r, ovzas z6 ^ «pos ro E' leya, 
ori iazlv rog ro A «p6s ro P, ovzais z6 yJ tcqos ro Z. 

Etlijq>9ta xwv \itv A, B, ti i&dxis itokKaaXdat 
ta H, &, K, Twv Sh F, E, Z alXa, « etvxBv, lam 
nokkankaaiix ta A, M, N. 

10 Kal ind iadxis iszl jroAAajrAaOtK ta H, zmv 
A, B, za 6i f^^p^ zots aSavzas itoXXaTiXaeioig tov 
ttvzbv B%Bi Xoyov, iaziv ttQa tog t6 A xffbg tb B, ov- 
zas zb H arpog z» &. dta ra avza 6ij xal as tb E 
jrpos t6 Z, ovzas zb M zpbg zb N' xtti iariv ag rb 

16 A irpog t6 B, ovtag t6 E nifbs t6 Z' xal d>s apa 
ro H aQog tb &, ouros t6 M «poff ro N. xal inei 
iativ ag zb B repos ro F, ovzas zb A xpbs zb E, 
xal ivaXXa% d>s zi B Ttgbg to ^, ovtas zo F Jrp6s 
TO E. xal intl ra &, K rmv B, ^ iadxis iszl TtoX- 

20 XaaXaaia, za 6h ^iQi} zots iedxis noXXuTtXaaiois zbv 
avzbv i%Ei Xoyov, Hetiv dpa wg ro B repog to ^, ov- 
rrog ro @ srpog ro K. uXX' d>s rb B Jtp6g to .^, ov- 
irog to -T jrpog t6 E- xal ag dga zb & xqos zb K, 
oCircjg ro F itgbg zb E. ndXiv, insl rd A, M rav 

25 r, E iadxi.s iatt, noXXajtXdaia , teziv aqa ms zb T 
TtQos zb E, ourros ro A «qos zb M. dXX' mg tb 



bZ. 



'1 



2. Snpra Iv add. v.ai m. S F. 3, TSwtttpayfteVii P, 

cort. 7. J] B cort. p. 8. a. fTujjfj'] mg. m. 2 post L 

nam 5 litt. F. 10. H] poat ras. 1 litt.T. 12. Ko^iurii _. 

i*. oBims] y.ai B; Om. p. 15. outo>B] om BVp. Poat 

hoc nerbum rep. P lin. 13: lo H — 16; to B. 16. ovx»%\ 



A\ — I Bi-H r\ — I 

d\ — I Ei 1 Zi — I 

H\ 1 — I 1 9 \-\ I 1 A\ — \ — I 

K\ — I — I — I • M\ 1 1 iVi— i-H 



ELEMENTORUM LIBER V. 65 

Sint tres magnitudines A, 5, F et aliae iis numero 

aequales binae si- 
mul coniunctae in 

eadem propor- 

tione J, E, Z, et 

perturbata sit 

earum proportio, ita ut sit ^ : 5 = JS : Z, et 5 : JT = -<^ : JS 
[def. 18]. dico, esse A: r= J : Z, 

sumantur magnitudinum A, B, ^d aeque multiplices 
H, ®, K et magnitudinum F, E, Z aliae quaeuis aeque 
multiplices A, M, N. et quoniam H, & magnitudi- 
num A, B aeque multiplices sunt^ partes autem et aeque 
miiltiplices eandem rationem habent^ erit A:B = H:® 
[prop. XV]. eadem de causa erit E:Z = M: N. et 
A:B = E:Z. itaque etiam H:S = M:N [prop. XI]. 
et quoniam B : F = A : E, etiam permutando erit 
B: ^ = F: E [prop. XVI]. et quoniam ®, K magni- 
tudinum B, A aeque multiplices sunt^ partes autem et 
aeque multiplices eandem rationem habent^ erit 

B:A = ®:K [prop. XV]. 
uerum est B: A = T : E. itaque etiam 

®:K = r:E [prop. XI]. 
rursus quoniam A^ M magnitudinum F, E aeque mul- 
tiplices sunt, erit F: E = A : M [prop. XV]. uerum 

om. BFVp^ 17. ovtoos} om. BFVp. 18. Post E add.^ xal 
sUrjntai tmv (ilv B, J laatiig noXXanXdcia ta @, K tmv dh 
r, E aAAof, a ^tvxBv, ladntg noXXanXdoia td A^ M^ ictiv 
aQU mg to ® nQog to A, ovtcog to K nQog tb M Bp et V 
mg. m. 2. 18. <ag] om. F. B] seq. ras. 3 litt. F. ov- 
t(og] om. BFVp. 19. B, J] in ras. p. 21. ovt(og] om. 
•FV. 22. ovt(og]^om. BFVp. 23. mg aQa to B] in ras. 
m. 2 V. 24. ovroDs] om. BFVp. 26. ovtong] om. F. 

Euolides, edd. Heiberg et Menge. n. 5 



66 STOIXEIfiN *'. I 

^rpdg ro E, ovtiag ro & UQog to K' xal ag «pa vb 
& npog lo K, oiittos t6 vi jrpog To M, xai EvaAA&^ 
tag t6 & 3tp6s t6 j4, t6 K nQog t6 M. iSeCx^ ^^ 
xal dtq t6 H ngog ro @, otjtos t6 M 3rp6s ro iV". 
5 ijitl ovv XQia (isyd^t} ietl ta H, &, A, xal aXia 
avtols Fea t6 jrA^&og r« K, M, N GvvSvo Aafi^avo- 
iieva iv ra «vrffl Xoyo}, xai iariv avxmv TetaQayft^viq 
rj avaXoyCa, Si' ("ffou ap«, s^ vnt^i%Et t6 H tow A, 
VTtEQixei xal to K rou N, xal fl taov, Faov, xal eI 

10 eAaTTOi', ^Aarrot'. X4icl isri r« fihi H, K tav A, 
ladxis aoXXaitXaeia, t« S\ A, N tav F, Z. ^tiv a^ 
tas To A itQog ro f, ouTCJg t6 ^ 3ip6j ro 

'Eav aqa f/ XQia (leyi&jj xal aXXa avrotg taa zii 
itXij&os ffwd^uo XaitfiavofiEva iv rp avta Xoy^, ]j di 

15 tstapayii^vr} avtdv ij avaXoyia, xal dt' taov iv t^- 
avta Xoya istaf offEf idii Set^ai. 



eI 

4 



x&'. 
'Eav iCffcotov «pog SevTspov rov avtov S^\ 
Xoyov xal tpitov apoq titaQtov, e%ti 3i 
ao xi(tnTov ffpog StvreQov zov avtov Xoyov xat 
exzov xgbg titaQtov, xal avvte&iv Ttgmtov xal 
3tifi3ctov stpog devtEQOv tbv aur6v ^^si Xoj^ov 
xal zffCtov xal £»roi/ WQos tira^rov. 



1 



2. ouinis] om. BFVp. Hic qnoque nOBnulla 
euanuerunt, ut legi non poesint. *. mo/] aupra V. ov- 

rws] om. BPTp. 6. ietXv avaloyav Theou (BFVp). ^Jite] 
supra F. T. Ante iv m. S insert. nai F, ia quo hic nonnalla 
Buetulit reBarciuatio. 8. 17] oni. P, 10. llaaaov, Haeaaf 
BVp._ 11, A, N TOT r, Z] in mg, tranBennt m. l, aeq. iu 
mg, (clla S fivjcv ^oanig, dein in teitu xoXlanXaeia F; 



I 



ELEMENTOEUM LIBER V. g7 



rr-. E = &: K. quare etiam &:K=A'.M [prop. XT], 
et permutando [prop. SVI] ® : A = K: M. sed de- 
Oionstrataiii eet, ease etiam H :& = M: N, iam quo- 
niam datae sunt tres magnitudiDes H, &, A et aliae 
iis numero aequales K, M, N, binae simul coniunctae 
iu eadem proportione, et perturbata eat earum pro- 
portio [def. 18], es aequo, ai H magnitudinem A 
Buperat, etiam K magnitudinem A^ superat, et si ae- 

»qnalia, aequalia est, et si minor, minor [prop. XSI]. 
tA H, K magnitudinum A, ^ aeque multiplices sunt, 
ji, N autem magnitudimim F, Z. itaque A: r= ^ :Z 
[de£ 5]. 

Ergo si datae sunt tres magnitudines et aliae iia 
numero aequalee, binae simul coniunctae in eadem 
proportione, et perturbata est earum proportio, etiam 
ex aequo in «adem propoilione erunt; quod erat de- 
moustrandum. 

XSIV. 
8i prima ad aecundam eandem rationem habet 
ac tertia ad quartam, et etiam quinta ad secundam 
eandem rationem habet ac sexta ad quartam, etiam 
compositae prima et quinta ad secundam eondem 
zationem habebunt ac tertia sextaque ad quartam. 



I 



68 ZTOIXEIQN e'. ■ 

'TZptorov yap ro AB npbg dsvtsQov to F rov ai- 
TOi' ixBra Xoyov xal rgirav zb -dE «pog zirctfftov to 
Z, ix^'^'^ ^^ ""^ xffiTtTov TO BH Kpo? StuTSQOv To r 
zbv ttvtbv Xoyov xal sktqv to E® jrpog r^T«proi' t6 
5 Z" AeVci "i^' ""^ fltwt£'9^i' ZQintov xal nifimov tb 
AH Kpog d£vTtpov ro F rov avTov f^n Xoyov, xal 
TpiTov xai «(Tov To ^0 ffpog TtVttprov To Z. 

'^ei yftp i<STiv tos to Bf/ npoj to F, ovrras ro 
£0 Jtpos To Z, KVUTiakiv a'p« tog to F Jtpog to BH^ 

10 ovTfDg lo Z JTpos t6 E&. ixEi ovv itSTiv wg ro AB 
agbe to r, ovros tb ^E irpos lo Z, a>g d\ to F 
apos To Bi/, ovTog to Z jrpos t6 E&, di ftfov apa 
iffrlv d>g zb AB Jtpos to BH, ovTmg t6 -^£ iipos to 
E®. Kal inel Sijj^ijfiivtc (ityi&Tj avdXoyov istiv, xal 

15 avvte&ivta avaXoyov Setaf ^ativ ctQa dig ti 

jTpog t6 HB, oiircjs To A& Jip6s t6 &E. Scti. SJA 
xal a>g TO BH Jtpog to J", oiiTtos tb E& itQbg i 
bC isov oQct iatlv (6g r6 AH jrpog ro F, ovrmg i 
^& repog ro Z. 

SO 'Eav apa nptoroi' Jcpos SsvtEQOV rov avtbv ij^f^ 
Xoyov xttl TQitov Jtpos TiraQtov, ^x?! ^^ '"'^ ^iykTttov 
jrp6s ^cvrEpoi' rov avtbv Aoyov xai sxtoi' jrpog t/rap- 
Tow, «al ffiwTEd-iv jrpmtoi/ xci itiiiittov jrpos SeiJTspor 
rov kvt6i' J^|£t Ad^j^ov xal tqitov xal fxTOf 

2S zdtaQtoV ojiBQ iSit Ssi^ttt. 



'Eav tisaaga fisyiQ-t] avaXoyov ij, T( (i. 
yitSTov [avTcavl xal ro iJla;(t«Tov dvo tav Xo 
xwv (isi^ovd iativ. 



ELEMENTORUM LIBER V. 69 

S Sit enim AB: r= ^IE : Z, 

"' '^ et Bff;r= £@:Z. dico, esse 

' etiam AH:r= ^@:Z. 

. 1 — — \B nam quoniam est RU : T 

— ( = E® ; Z, e contrario erit 

[ [prop. Vn coroll.] T : BH = Z : £®. iam quo- 
ist AB:r= JE:Z, et r:BH=Z:E&, ex 
I aeqao erit AB\BH = ^E:E& [prop. XSII]. et 
L >guoniam diremptae magnitudines proportionales simt, 
1 etiam compositae proportionalea enint [prop. XVIII]. 
itaque AH ; HB = ^0 : @E. uerum etiam 
BH:r=E&:Z. 
I itaque ex aequo AH : F = ^® : Z [prop. X5II]. 

Ei^o si prima ad aecundam eandem rationem habet 

ac tertia ad quartam, et etiam quinta ad aecundam 

eandem rafcionem habet ac sexta ad quartam, etiam 

compoaitae prima et quinta ad secundam eandem 

. iratiouem habebunt ae tertia sextaque ad quartam^ 

I gnod erat demonatrandum. 

xsv. 

quattuor magnitudines proportionales sunt, 
llQ&zima et minimB duabus reliquis maiorea suut. 



XXV. Kntociiu id ApoUon. 



. 139. 



1, fiii" yap P, 6. lO npmrov FV. aiilMiOV x6 AH^ 

xtfi (es Tiai) xiiittzov, lo jiH Bapru tp. 8. kuI intl yuf F, 
I Mai del. foti F. 12. aqa] snpra F. 1*. imiv] PF,- comp. 
- ^,T T>. .:, i„- Theou? (BFVp). 
- 36: Stkat' ' 



ipra F. 
15. fffrtv aQa (dc] F; b. 
16. HB] BHP. iiniv B. 21. ?xv 



a lotna p. 
loitiii B. ^o. 
tit 3vo P9F, et 



ii.iX" 



'.. KCll l. 

L. P, Eutoeiae. 



- 25: 3i£^tn] nai t 
Svo] Eutocins, V 
:mv] om g;. 



70 ETOlXEmN t 



r^Eiftai t^aaaQa (leyB&f) avaXoyov xu AB, ^^» 
£, Z, tos Tu -^-B tpos Eo jT^, oStojs ro E iiQoq ro 
Z, ^ffTo 6% ftiyiSzov (ihv ttvtmv to AB, ika%Lgvov tfi 
10 Z' kiya^ oti ta AB, Z tmv FA, E (iti^ovd invi^ 
5 Keie&ta yap rm ftiv E teov to ^lf, rp S\ 
laov 10 r©. 
'£«si [ovv] ietiv rag to >fB tcqos i:o TA, ovzm{ 
To E Ttffog to Z, fffov 81 th fiif E t© AH^ t6 dfr] 
Z Tw r&, SifTiv Kp« (6e To AB Ttpog ro FA, omatg 
10 i6 AH jcgog to F®. xal ixsi isttv mg oXov ro 
AB TCQog oi.ov t6 FA, owira? a(paiQE&£v to AH itffQg 
«fpaLQE&^v t6 r&, jtal lotnov apa ro HB «pog Xoi- 
xhv ro 0.^ iatai ag oAoi/ ro ^B jrpoj oXov t6 F.^. 
ftcf^ov 6h t6 AB Toti J"^' (tBi^ov aga xctl t6 HB 
5 Toti &A. xal inel ftfov ietl to [liv AH ta E, t6 
dh r& tip Z, ta Spa ^H, Z fffa ^flri rotff T®, E 
Kal [inelj iav [aviaoig lect irpoffrEd^, ta oXa «vmA 
iaxtv, iav apa] riav HR, ®A avCicav ovtmv xal (lei- 
lovog rou HB tm ftev HB aQoOtsd^ ta AH, Z, t^ 
20 5i @z/ 3rpo5T£^^ ra r&, E, evvdystai zd AB 
[isitova rrav FA, E. 

'Bdv aQa TEffffapa (leyi&i] dvdXoyov i,, tb /tiyt 
Ozov avzmv xal t6 iXd%i,6zov Svo zmv Xotxmv fuitovi 
iaziv omp iSei tf£f|at. 

2. E] (alt) 9 Tt. 4, iaxtv^ PF; comp. p; lo-n BT.. 

5, T^l 10 V qi (non F). To) ira Vip. r^] lO V 
To] %m Vi om. P. 7. o«»] om. P'. 8. Z] in ras. nr 

13. ri9] 8 e eon, V. Poat W 2 titt. euan. F, 
^B «. 13. 0J] A etftB. F. I^oiat] aeq. ras. P, i 
fmttt inB. qi. .iB] B e corr. F. 16. ^ H] H oort 
V m, 3. IS. *!?] m. rec. p. ^H] P, BH », ^.ff 9, 
3iu] BUpca m. 1 V. 19, ti] rd Vj oorr. m, 2. 
m. 2 V, 21, liliitova q). 22. aqa] om. p. uvaloyov 



li 

1 



Z 

I 



J 



I 

I 



ELEMENTORITM LIBER V. 71 

^ _ Sint quattuor magnitudiHes 

proportionales ^B, FjJ, E, Z, 
^ ita ut sit AB : r^ = E : Z, 

' — 1^ et maxima earum sit AB, mi- 

— ■ — -I nima autem Z. dico, esse 

AB-\-Z>r^ + E. 
ponatur enim ^H = £ et r& = Z.') iam quo- 
niam est AB:r^ •=- E:Z, et £ = AH, Z = F®, 
•rit AB : P^ •= AH : r&. et quoniam est 

AB:rj = AH: FS, 
«it Itiam [prop. XLK] HB:€>J= AB: VA. sed 
AB > rA. quare etiam HB > &A?) et quoniam 
^H=£: et r® = Z, erit ^H + Z = r@+£. et 
si datia ma^itudimbus HB, ®A inaequalibus, qua- 
rum maior est HB, magnitudini HB adiicitur AH-\- Z, 
'z/ autem magnitudini maguitudo F® -}- E, couclnditur 

^B + z > r^ -I- £.») 

1) Nam cura AB > E, erit TJ > Z (prop. 14). 

2) Cum HB : Sj3 — AB : TJ, erit {prop. 16) AB : HB 
. rj : « J; tnm n. prop. 14. 

3) Cum I xoiv. fvv. 4 Bubditiua Bit, uetlia Imi et qv^- 
«ij — iav ai/a lin. 17—18 necesBario delenda aunt, praa- 
sertiin cum b3.ec poBtulati forma &d demouBtrandum propo- 
Bitiim non aufficiat, et oftendat orationia forma ob repetittun 
iiv psmoleata; ad qnam molestiajn leuaadam iKCi lin. 17 
siiatulit AugustuB. sed fortaaae Euclidee ipae lin. 17 sq, 
haec BoU scripaerat: £azc za A B, Zzmv FJ, £ fiF^fova iativ; 
nam ovvayezut lin. 20 inusitatnm est de demonatratioue, qua 
vSa. poterat Euolidee, cfr. uoL I p. 181 not. 



Minu a. 23. llajtOTO»] Hoim» V. In fii 



^L «XDIIE^D) 





■Bpo.. 
a . 'DftotK G%'i\^a.Ta Bv&vy(ftt(i(ia iariv, 
Tff? re yfavias £oas ixn xara niav xal ras ati/l x^ 
taas ycoviag jzlavpag dvdXoyov. 
5 [/5'. 'j^vrtTttnov&^ora dh ffjjiJftaTa ieriv, o^tav i 
ixtttd^of xav axiiiidtoiv riyovfuvoi te xal atoiieva 
Aoyot loetv.] 

y'. "j4xqov xal (tioov Koyov ev^eta CETftn 

a&ai i.iyBTat, ozav ^ ms il o^ij irpo? ro ftfiE§(» 

10 Tfiijfia, ovrtos t6 [lettov ngog zb iXaxrav, 

3'. "IVoff iarl reavTOff sj;i}fL«tos ^ cjro T^g xo^9^ 
inl r^v ^daiv xd9£Tog dyofidvi]. 

[e'. Aoyag ix loyoiv 6vyxsla&ai i,iyErai, orav «W 
rmv koyav Jtijitxoiijrsg igi' iavtds TtoXKttTtXaaiaa&^eJ^l 
15 aat notiaai rtvtt.^ 



Td TQiyava xal Tci ■naffttkX^KoyqaiHta « 



Def. 1. Heco def. IIB, 1. 2. Hero def. 118, 1. 4. Cfr. 
Heco def. 73. [5. TheoD in Rolem, I p, 235 ed. Halma. 
EutocinB in Arohim III p. 140, 23. Bftrlaam logist, V def. S]. 
Prop. I. Proclns p. 246, 6. 405. 11. Pappns V p. 483, f" 
VIlI p. 1106, 23. 



■ loyo'] P, P Bnpra acr. , 
opoi Bp et V in ra»., eupra scr. loyoi ta. 2; l^iav Z^oi Caa.»^ 
dalla, PejT-ardus; iDyoi iam Hero. tlmv F, wai p. Deiii ae4..J 




VL 

Definitiones. 

I. Figurae rectilineae similes sunt; quaecunque et 
angulos singulos aequales habent et latera aequales 
angulos comprehendentia proportionalia. 

[II. Beciprocae autem figurae sunt^ ubi in utraque 
figura et praecedentes et sequentes rationes sunt].^) 

ni. Secundum extremam ac mediam rationem recta 
linea secari dicitur^ ubi tota ad partem maiorem ean- 
dem rationem habet ac maior pars ad minorem. 

IV. Cuiusuis figurae altitudo est recta a uertice ad 
basim perpendicularis ducta.*) 

I. 
Trianguli et parallelogramma sub eadem altitudine 
posita eandem inter se rationem habent ac bases. 

1) Haec definitio nusquam ab Enclide nsurpatur; neque 
enim ad illustrandam locutionem Xoyov dvxinsnovd^ota iv^iv 
aut opus est, aut, si opus esset, sufficeret. praeterea Zoy ot 
lin. 7 obscurum est. itaque puto, Simsonum p. 870 iure eam 
damnasse. fortasse ex Herone sumpta est, apud quem legitur. 

2) Def. 4 om. Campanus. Def. 5 sine dubio interpolata 
est; nam nusquam usurpatur nec apud Campanum exstat neque 
in ipsis codd. locum eundem obtinet. sed cum P a manu 
prima addito signo, quo in textum referatur, eam in mg. 
nabeat, fortasse ante Tbeonem interpolata est. u. Simson 
p. 372 sq. 

def. 6 in Bp. 9. 17] om. PBp. to] om. F. 10. iXaaaov 
FV. 13 — 16. mg. m. 1 P; om. hoc loco Bp. 17. xd] (alt.) 
supra m. 1 F. 



lOV, 



74 STOIXEiaN S'. 

vito t6 avtb viliog ovta jrpos aXXfjXu iartv ag 

"Efltra t^iyatva (isv ta ABF, AF^, 3tapaiA»)Ad- 

y^Aft^a tfi T« iiT, fZ iiwo t6 avto li^os ro ^T* 

6 kiya>, oTt ^UTiv los i/ BF ^KtHg wpog t^v Pz/ ^aOiV, 

Qvtcas tb ABF tQCytavov nrpos ro AV^ Tptj^ojvov. 

xal t6 EF ■sta^aXXrilQyQa^^ov mpos to PZ «api 

'Ex^E^l^ed-at yag ^ S.-i iqj* ExaTf^tt tk ^^( 

10 ial ta @, ^ er]ii£ta, xal xtCa&aeav r^ /iW BT §t 
aei Caai [offaidijreoTotiv] aC Bli, H&, zy Si FJ ^o- 
eu taai oaaidfiaorovv at ^K,Kji, xal iae^av^^^to- 
aav at AH, A@, AK, AA. 

Ka\ iitBi iaai sCelv at FB, BH, H& aAA^JUos, 

15 ffla iatl Xttl ra A&H, AHB, ABP tQCymva aUs}- 
Aoig. baaitXaaCav aga iozlv ^ ®r ^dsis trjg BV 
pdaetos, xoaavraitXdGLov iati xal t6 A&F t(fCyavov 
tov ABr TQtycavov. Sia Ta auta di] oaaxXaaCav 
itlTlv ij AF ^asig t^s VA ^a6Ea>g, Toaavttatldeiov 

30 ieti xal t6 AAF TQiyatvov row AP^ Tgiyavov xal 
al l'at] iaTiv 71 &r pdeig tfi FA §daEi, leov ietl 
xttl to A&r tpCyavov Tta AFA T^ijfojvfo, xal el 
vatQixEi. ^ &r ^deig T^g FA fidaEtog, vitEQiy^Bi ««i 
t6 A&r TQCyavov toi AFA tQiyavov, xal si iXda- 

25 aiav , IXaaeov. tEaaaQiav St] ovTav (lEyE&^av Svo 
[lev fideEav tav BF, F^, Svo Sl TQiyavav Tmv jtBP^ 
AFid ECXtjJtzat ledxig noXXanXdeia Tijg (ilv BF pd- 
(Jicas xal Tov ABV TQiymvov ^ te ©f ^daig xal t6 



4. rz] Z e 
(BVp, F in tai 



ovttc Theottfl 

11 



[ 



ELEMENTORUM LIBEK VI. 



76 




A 



Sist triaDguli ASr, AV^, pa.rallelograinma autem 
Er, rz aub eadem alti- 
tudine poaita AF. dico, 
esse Br : T^ = ABF 

•.Ar^ = Er:rz. 

producatur enim S ^ in 
utramque partem ad punc- 
ta &, A, et ponantur basi BF aequales quotlibet rectae 
BH, H& et basi Tz/ aequales quotlibet rectae AK' 
£A, et ducantur AH, A&, AK, AA. 
' et quoniam TB = BH= H@, erit etiam 

A A&H = AHB = ABF [1, 38]. 
itaque quoties multiplea est basia &r basis BF, to- 
tiea multiplex eat etiam triangulus A&P trianguli 
ABT. eadem de canaa, quotiea multiplex eat baais 
AF basis FA, toties multiplex est etiam trian- 
gulus AAT triangiili ATA. et si @r = TA, erit 
etiam A A&r= AFA [I, 38], et si ®r> VA, erit 
etiam A A&r> AVA, et si @r<rA, erit 
£\ A®r < AFA. itaque datis quattuor magnitudini- 
)UB, duabus basibus BF, F^ et duobus triangulis 
Br, AFA sumptae sunt aeque muliplices basis BF 



t^v B^ y:a&izov ayonirtiv Theon (B^p, F in laa. ui. 2); Btid 
cfr. (ief. 4. 5. l/yiD, oii] in raa. m. 2 F, Imlv los ^ B ri 
in DiK' transeunt m. 1 F. paotg] -i; in mB. F. S. S^J 

^B Bp, Vm. 2. 11. oauiSri«oioiv] ora. P. 12. J K] 

in m. V. 11. BH, H8] e corr. p. 16. ^atiV P; comp. p. 
AUe Fp. 18. JBF] corr. ex .rfSr m. 2 F. 19. ^r] 

r^ P, Hfld .4 inrafl. TJ] JTBp. 20. .,<r.J] ^jrBp. 
Vr'»»"™ » (noD P). Bl, r./il inter T et ^ raa. 1 litt PV, 
itniv P, comp. p. 22. AAFBp. 23. VA] inter T et j1 
rB8. 1 litt. V. 24. AFA] PV, B in ras. m, 1; AAP p; 

j*Br F. Wtftto» Hartov BF (^IoTtoiv F in. 2). 







76 rroixEiiiN <s'. 

A&r TQlytovov, T^? S\ r^ ^Bttsias xal xov 
Tffiymvov ttXXtt, a hv%tv, leaxig noXlaiiXaeia ij ri 
jir ^asig xal to AviT tgiyavov xal S^$Bixttti, oti. 
bI VTCBQixBi. ij &r ^aets t^g PA ^cffEOg, vnB^ixei 
5 xctl t6 A@r tpiycavov TotJ AAF tgiytavov, xal sl 
teri, ieov, xal ti iXdeeav, iXaeeov ietiv aga ras ^ 
BT ^aotg irpog t^i> F^ ^daiv, oOtrog to ^Bj 
Tpiyavov npog to AFjJ rgiyavov.' 

Kttl inel tow ftfi' ABT tffiymvov dmXdeiov 

10 To Er attgaXXTjX6yQa[iiJ.ov , lou tfJ AFA TQtymvov 
diJtXdetov ieti ti Zr naQaXXriX6yQa(i(iov , tcc tfi 
jiEpij Tofg toffKtJTrag itoXXaTtlaeioig tov avtov i%Ei 
Xoyov, iariv aQtt rag t6 ABF tgiyavov xpog ro 
^r^ rp/ytjvov, oi^tms ro £F jrapaAAi;Aoypaftjtov 

15 Wpog t6 Zr 3tttpaAAijAn'y(iKf(|Uov. fJTfi ovv iSsCx&f], 
toj fiiv ^ BV ^deig irpos t^v T^, ovtag to ^BJ" 
Tptj^toi/ov ^rpos t6 AF^ tgiyavov, ag di to ARF 
tQiyavov Jipog lo AVA TQiyavov, ovtag ta EV 
itaQ«XkriX6yQa[iyiOv Jtpog r6 TZ ^rapccAATiAo^paft.fioi', 

20 xal d>g ap« ij BV ^deig itQog tijv VA ^deiv, ovtag 
tb EV xttpaXl7iX6yQaji[iov irpo? t6 ZV nupttXXijiO' 
yQanfiov. 

Ta KQa tQiywvtt xal tk aaQaXXrjXoyQaiiiia ta 
vita t6 cut6 \iii>og ovta rcpog akXjiXd iativ mg 

35 ^ttOeis' oasQ SSsi Sei^ai, 



i 



'Ettv tptytovoy jrapa (liav tiav xXbvq 
dx^fj tig f^fl-ffa, avaAoyou te^El Tag rotJ 

2. 5] aupra F. S. AVirA P, 4. FA] A in «b. 

m. 2 P; AP F. 6. iiii)] taovB, etF, oorr. m, 2. Iluffocggr] 



ELEMENTORUM LIBER VI. 77 1 

triangulique ^BF basis @r et triangulus A®r, et 
basis r^ triangulique AdF aliae quaeuis aeque multi- 
plices basis ^F et triangulus AAF] et demonstratum 
est^ si @F basis basim FA superet, etiam triangulum 
ASr triangulum AAF superare, et si aequalis sit, 
aequalem esse, et si minor, minorem. itaque erit 

Br:rA = ABriAFA [V def. 5]. 
et quoniam Er=2 ABF et Zr= ^AFA [I, 34], et 
partes eandem rationem habent atque aeque multi- 
plices [V, 15], erit A ABF : AFA = EF : ZF. iam 
quoniam demonstratum est, esse 

Br:rA = ABF : AFA 
et ABF^AFA = EF: TZ, erit etiam 

Br:rA = Er:zr [V, ii]. 

Ergo trianguli et parallelogramma sub eadem alti- 
tudine posita eand6m inter se rationem habent ac 
bases; quod erat demonstrandum. 

11. 

Si in triangulo uni laterum parallela ducitur recta, 
latera trianguli proportionaliter secabit; et si latera 



II. Schol. in Archim. III p. 383. 



iXaGGov P; iXaxxov B, et F, corr. m. 2; ^Xarroov p. IXarrov 
BFp. 9. il\v xov V. 10. ds] m. 2 V. 11. loxiv P; 
comp. p. 12. noXXanXaaCoig] naqanXricCoiq B; corr. m. 2. 

16. Zr] rZ BFp, V m. 2. 16. 17 /*aV p. ^^^ 

ATB P. 17. ATJ] corr. ex AJT F, xQCymvov] om. V. 
18. xQCycavov'] om. V. A TJ] e corr. F. xQCymvov] m. 2 V. 

19. rZ] P, y m. 1; Zr BFp, V m. 2. 20. Pz/] ^Tp. 

21. naQdXXriXoyQayiyi^ov] (alt.) om. V. 27. naQo. yLCav] mn- 
tat. in naQaXXriXoq iiia B m. recentissima ; in V supra scr. 
m. 2: ^rot (iia xmv nXsvQoav naQaXX^qXog. 




,78 ETOIXEIiiN 5'. 

ytovov aisvQag' xal ictv et rou TptyojV 
pal avaXoyov TfiTj&aifiv, rj izl zas tro/tag 
init,Evyvv}iiv7} 8v&tta nu<fa rijv i.otni}v iatai 
tov Tffiycovov TCXsvQav. 
5 Tfftyfovov yaff tov ABT nu^af.ky}Xos fli^ tAv 

nKevQtov ttj BF ^j;*o ij <JE' Xiyco, oti iatlv is ^ 
B^ Jipog TTjv /tA, ovTMg ^ rs xpig tijv EA. ^k 
'Enei,tv%&aaav ya^ aC BE, VA. fl 

"laov a^a iatl ro BzlE ZQlymvov ta V^iE rpul 

10 ydvqf iml ya^ tijs avt^g ^aSedg isti. tijq ^E xal 
iv tats avtatg irapaAA^Aoig tatg ^E, BF- «Wo Si 
ti ro A^E rgiytovov. ta 6i laa npog ro awro ■eov 
aurov SxEt loyoV iaziv apa wg ro Bi^E tifiycavov 
apbs tit A^E [rptywvoi'], ovrog ro r.J E tQiyatvov 

16 Kpog 10 AjdE Tgiyavov, alf.' rog fth/ ro BjdE 
tQiymvov Jtpos to AjJE, ovtats ^ BA srpog t^v 
AA' vab yuQ tb avtb vijios ovra tiiv amb roij E 
iai tiiv AB xafrETOv ayofiivrjv wpos aXXijiit sitfiv 
tog at ^affftg. Sta ta avta Sii cos tb TAE tffiyojvov 

ao JEpog lo AAE, oureg i\ FE ;rpos tijv EA' xal 
(og apa ^ B^ repog zi}v jJA, ovrtgg t} VE xgt 
tijv EA. 

'AXla Si] at tov ABF tgtyavov itKsvQaX at AB 
AF avakoyov rfTfnJflS-toffav, mg 7) BA Jtpog zijv jdA 

25 ovrrog ij FE XQbg tijv EA, xal ias^ivx-^ei ij ^^ 
Kiym, ozi napUilAijAog istiv ij ^E r;; BF, 

Tmv yaQ' «vtdov xaraaxtvaa&ivtav , inai ifltivl 



1. Aute iav 2 litt. eras. V, &. notpu t^v lotn^] mDtbt. 

ia 'nafallriXoi tji iQi-aji B m. recentiBs ; in F supm KT. iHl 3 

«oetellijlos. 4. iil*vp«i»>] mutat. ie xlivfa m. recentJBS. B. 

7. i^v] pOGte& inHert. tp. iiji'] pOBtea inBert, tp. EA} 



^^ 



ELEMENTORUM LIBER VT. 79 

trianguli proportioiialiter secantur, recta ad puncta 

sectiouum ducta reliquo lateri . triauguli parallela erit. 

Nam in triangulo ABF unj 

latenim BP parallela ducatur 

^E. dico, ease 

Xj jF B.d:^A = rE: EA. 

N^^^^^^ dueantur enim BE, Fd. 

itaque A B^E = r^ E\ nam 

in eadem baai sunt ^E ei in iisdem parallelis jdE, 

Sr [I, 38]. alia autem quaedam magnitudo est 

A AjJE. et aequalia ad idem eandem rationem ba- 

bent [V, 7]. erit igitur BJE -.A.dE-^ r^E ; A^E. 

uerum BzJE : A.JE = B^: jJA; nam cum sub eadem 

altitudine positi sint, ea quac ab £ ad AB perpen- 

dicularia ducitur, eaudem inter se rationem babent ac 

Iwses [prop. I]. eadem de causa erit etiam 

^ r^E : A^E = TE : EA. 
quare etiam B^ : ^A = FE-.EA [V. U]. 

iam uero trianguli ABF latera AB, AF propor- 
tionaliter eecentur, ita ut ait B^ : JA = FE : EA, et 
jdocatur j^E. dico, .dE rectae BF parallelom esse. 



' AB F. 8. yaf] Hupra m. I V. 9. opa] Sij P. MiV P, 
oomp. p. 11. Br] EZ <p (dod V). 14. -co] corr. ei 
tp m. 2 T. ^JEJ JAE P. xeiyavov] om. P. ti/i- 

yuvov] om. V. IG. AJEJ J e corr. m. 2 V. ;j] ip; 
add. »apra etiam m. rec. 19. Post Baaeit a.dd. V; tai ii 
to rjE itgof ro AJE teiymyov, i^] om. F; uidetui add. 
fbbse m. 2, sed enan.; Sfj xai P. as 16} om. V; As 8i 

td ip. rJE xtiyiavov npos id AJE] om. V. 30. EA] 

JE p. 21. TE] TB P? 23. a! AB, AT] m. 2 V; bT 

om. F, add. m. 2i. Ante ms hab, Bp: xhtu td .f/, E 

OTjiictd\ idem P mg. m. 3. at Sfa Bp. 25. VE] mntat. 
in Br m. 2 7. 



I 



I 




80 STOIXElfiS 5'. 

ms ^ B^ itQog T^v ^^, ovtiag i} FE jrpos v^vi 
aXl' d>s fiiv rj Bi^ Ttgog rrjv ^A, ovtoiff ro B^E 
TQCyavov repog ro AAE xglywvov, dtg dh r) FE agog 
ti}v EA, ovtag to F^E tQiyoavov rcpos to A^E 
6 TQCyowov, xbI cog «pa to B/IE rpCytovov «pos w 
ji/IE tQ(ya>vov, ovr<ag to P^ E tpCyavov xgog ro 
AAE tQlytavov. sxdttQov aga ziov B^E, T^E 
tQiymvKiv TCQOS to AdE tov «■urov l%&i koyov. l60v 
aga iatl tb B-idE tQCyiovov ta PjdE tpiymve)' xaC 

10 bIoiv inl Tijs avtijs ^Ketcog tijs ^E. ra d% tSa 
tQCyavK xal inl rijs avf^g ^aaEag ovta xal iv taZs 
avtatg artQaXXriKoig ietCv. «aQakXrilos uQa iotlv 
^E t^ BT. 

'Eav aga rgiycovov naga ^Cav tmv JtXevgmv k%\ 

16 tts tv&ata, avdXoyov tBjist tas tov tgiymvov alEvgdg' 
xal iav at row TQiyavQV TtkEvgal dvdXoyov Tfttj&^metv, 
7} ijil tdg tofitts iTti^svyvvftivfj ev&sta itagd tijv Xot- 
Ttrjv sotai tov tQiycSvov atXevQKv' oiteg eSsi dst^Ui 



-gae' 
ietv, 
Xot- 

tia^m 



20 'Edv tQiyavov ij ycovCa SC%a iiiTi&fi, rj Si 
riftvovaa riiv yavCav ev&eta tiftvn xal tijv 
^deiv, td tijs ^dasag tn^^fiata tov aWToi' s^st 
Xoyov tatg i.oi.^ats row rQiycovov sXBvgatg' 
xal idv ttt r^s ^daemg tftrj^ata tov avtov 1%^} 

36 Xoyov tats XoiJtatg roi' tgiymvov nJtftipar^, 
17 dxo T^s X0Qvg>ijs i^i r^v tofiijv iai^evyvv- 
ftivri Ev&sia SCxa VEfiEt tijv toij TQtymvov 
ymvCav, 



3. rpiTyjomv] (alt.) om. V. 1. i^v E.J] to EA aeq, 
B, xoil ati S-na — 7: Ad^E. x^ltmv<iv\ mg. m. 



rifata 






ELEMENTORUM LIBEE VI. 81 



»nam iisdem cooiparatis quoniam est 
B^ :^A =rE:EA, et B^ : ^ A = /\BJ E: A ^ E, 
et VE : EA = A r^E ; AJE [prop. I], erit etiam 
A B^E : A^E = A r^E : A^E [V, 11]. itaque 
uterque triangulus B^E, FJE ad A^E eandem 
rfttioiiem habet. -itaque AB^E-^ T^E [V, 9]. et 
L jn eadem baai sunt ^ E. trianguli autem, qui aequales 
J 'Mmt et in eadem basi positi, etiam iii iisdem parallelis 
■-«ant p, 39]. itaque ^JE reotae BF parallela est, 

Ergo si iu triangulo imi laterum parallela ducitur 
I mcta, latera trianguli proportionaliter aecabit; et ai 
I ilatera trianguli proportionaliter secantur, recta ad 
T puncta sectionum ducta reliquo lateri trianguli paral- 
L-jLela erit; quod erat demoustrandum. 

in. 

Si angulus trianguli in duas partea aequales 
r'.diuiditur, et recta angulum secans etiam basim secat, 
partes basis eandem rationem habebunt ac reliqua 
latera trianguli; et si partes basis eandem rationem 
habent ac reliqua latera trianguli, recta a uertice 
ad punctum sectionis ducta ajigulum triauguli 
partes aequales secabit. 



1" 



IU. Theon in Ptolem. p. 201. Eutociua in Arohin 
272, 11. Schol. in Pappura III p, 1176, 16, 26 ftl. 

■.^i^aivov] (prina) om. BFVp. 7. TQiyai/ov} eomp. F. 8. 

«pos TO A^E] Bupra m. I F; n^oc to AJB r^imov V. 9. 
iexCv PV. 11. xal] (prjuB) ta F. 12. «oeaWjjioe V; corr. 
m. a. kiix£v] (priuE) PF V; latl B, et p (i in ras.); elaC V 
m. 2. 14. xltv^mv'] mg. m. 1 P. 20. Tf\ om. V. W&^] 
in raa. m. 2 V. Si] supra m. J P. 21. T^fi»n] xi^iin 
eros. I V. 24. Kcfl iav iq — 25: nUvfais} mg. m. 2 V. 24 



hv] eo^' ei ?it( ni.__l p. 27. wfisf] P. Fm. 2, V i 



I, «dd.Bi'it 



r 



82 STOrsEiiiN s'. H 

"Earca TQiyavov ro ^BF, xal TEZ(i^«9fo ^ vxo 
BAV ymvCtt Siy^B vxo t^s Aid cv&eiag' Kiyto, oTt 
sazlv ag ^ B^ «pos i^^v F^d, ovrmq 17 BA tc^os 
xijv AF. ~ 

5 "Tlj^d-ai yicff 3ia rov F t^ AA naQai.XnXoq ^ TB 
Ka\. diaxQ^eiau tj BA cvfiainTdrm «ur^ xara to 

Kal ijTfl tlg napaXXi^lovg rag AA, EP Evd-etii 
ivinmsv ij AF, rj aQa vno AFE yavia fflij iot 
Ttj vno FAA. kXX' 7i vao TA/i t^ vich BA^ i 

10 xtitai Han}' xal 73 vnb BAA aga t^ vno AFE iariv 
te^. xaliv, imel slg xapaXX-^Xovg ras AA, EF tv- 
&eta ivinEOEV 17 BAE, ij ixrbg ytavia ^ vico BA^ 
tatj iazl rri ivrbg rjj vjtb AET, iSsiji&T] Sl xal t] 
vitb AFE zfj vjcb BAA teij- xal i} vah AFE a(fa 

15 ymvia r^ vab AEF iariv ffftj- roffrc xal JcXEvpa ^ 
AE nXsvQa zij AF iaziv ffftj. xal inel zpiymvov 
Toii BFE naptt (liav tmv nXEVQiBv zijv EV ^xrat ij 
AA, avdXoyov a^a iarlv wg ij BA XQog riji' AT, 
ovtcag ^ BA Kpos T^v AE. ffffj di {j AE tij AV' 

20 mg aga ij BA wpog r^v AF, ovrtos jj BA nQog 
T^v AF. 

^AXXa dij /sto mg ^ BA npog T17V AT, ovroj 
7] BA npbs rijv AV, xal iat^vx&o^ V A^' ^dyt 
uTt dixcc ririiTjTat v} vno BAF yavia vxb njg AA 

25 Ev&eiag. 

TSv yip avtcov xataaxEvaa&ivTav , ixsi iot 
(OS 57 S^ srpog njv <::?r', ovzag rj BA TtQOS TTfv AS 
aXXa xal ms ^ BA jrpog tijv jJF, ovTog ^UTiv ^ 



1. KB^] BQprft F, 3, TiJ] <dr P. 7. ev^sttts VJ 

8. ^r^ntofl*] P»p Bp; {(nthnzemcv V, ^(«& P; CC " 

9, aUd P. 11. iv9iia\ tv&fias addito tv^rCa m ' 




ELEMENTORUM UCBER VI. 83 

« 

Sit triangulus ABF, et L BAF in duas partes 

aequales secetur recta A^. 
dico, esse 

B^ :r^r=. BA : AR 
ducatur enim per F rectae 
AA parallela FE^ et pro- 
ducta BA cum ea concurrat 
in E [I att. 5]. et quoniam in rectas parallelas A^^ 
^rrecta incidit AF, erit L AFE = FAA [I, 29]. sed 
supposuimus L PA^ = BA^. quare etiam L BAA 
= AFE. rursus quoniam in rectas parallelas^z/;£7F 
recta incidit BAE, erit L BAA = AEF exterior angu- 
lus interiori [I; 29]. demonstratum est autem, esse etiam 
L A FE = BA^- quare etiam L A FE = 'AEF. quare 
etiam AE = AF [ij 6]. et quoniam in triangulo 
BFE uni laterum EF parallela ducta est AA, erit 
BA : Ar = BA : AE [prop. II]. sed AE = AF. 
itaque erit 

BA:Ar=BA:Ar. 

iam uero sit BA : AF = BA : AF, et ducatur AA. 

dico, L BAF in duas partes aequales secari recta AA. 

nam iisdem comparatis quoniam est BA : AF 

= BA : AFj uerum etiam BA : Ar=BA:AE (nam 

2 V; svd^siccs svd^SLU Bp. 12. ivinscs V. BAE] litt E in 
rae. m. 2 P. 17] (tert.) in ras. V. 13. tarf] -ij e corr. m. 
2 P. ^^n litt. Er in ras. P. 14. BAJ jcorr. ex BJJ 
m. 1 p. aga ycovia] om. V. 16. AE] A0 n (non P), 

EA (p. nXsvgdv n (non P). 18. «^off %r{v] Tr{V comp. scrip- 
tom cnm 9rpo<r coaluit in F, nqo^ 9, et sic in seq. saepius. 

20. mq apa] P; Uxw aga mg Theon? (BPVp); cfr. p. 68, 16. 

22. BJ]J corr. p. JF] FJ P. 26. insl yaQ qf, 27. 
AF — p. 84, 1: ngos t?Jv] om. Bp. 28. ttJv] om. P (inser. 
m. rec, sed eras.). 

6* 



84 



ETOIXEISN S'. 



Xfibg t^v AE' TQiYfovov yap tov BTE mxQu fiiav 
rj)v Er 7jKtai ij j4jit' xal tog aga tj BA jcqos tijv 
AF, ovtag 71 BA repog x^v AE. fCij «9« t] AV ty 
A E' aOTi xal ymvia t/ uno A EV t^ vjto AVE 

h iattv foij. oAA' ^ (ikv vao AEV r^ ixxbs v^ tmo 
BA^ [ietiv} [071, 7} de vnb AVE tt} h^aUkl tj 
VTcb VAA iariv ftf*;' xal rj v^xb.BAjd aqa t^ 
VAA iextv £aii. ^ agaimb BAVy&via Slxa tir^y 
vnb r^s A^ evS^sCttS- 

D 'Ettv ccQa tpiywvov 17 ymvia 6(xa tftjjO^, 17 ii 
TB(ivovaa tijv ytoviav tv&ala cffwjj xal t^v ^Oiv, 
Ttt Tijg §a<Ssas tfi-^itatce roi' avtbv t%ei Ivyov tul^ 
lotaatg tow TQiymvov wAfiipars' xal ittv ta r^g ^dtleioe 
tfi^(iata xbv avxbv ixV ^oyov TBfs loiXutg zov rpi- 

5 yf^vov nlevQaig, ij axb r^g xoQvipijg inl t^v «Ofi^ 
imiivyvviiivT} ev&sia Si^a tijivti xiiv rotJ t^iymvt 
yatvCav' OXBQ SSsi Seti,at. 






Tav taoyavCav tpiytavav avdXoyo' 
V al nlBvffal at xsqI rag Laag yavCag xal 
.oXoyoL at vitb tag Haag ytavCag vTtozaivovaai. 
"Eatto iaoytovia tgCytova ra .ABV, j^VE Caiiv 



■tivm 



' filv vnb ABV ytovCav r^ vnb /JVE. tijv 
BAV t^ vnb V^E xal hi tijv vni AVB 
26 ty imb VEjd' i.iya, ott twv ABV, jdVE tQiyavtov 



Sfovta 1 
S\ 



IV. PseUuB p. 70. 



8. owwfi] m, 2 V. 
PBp;yte*(i<t«rV. 
^0«».] om. 1*. ^ Si' 
est in Vi AEVY. 



AB] AP q/. 4. AE] EA f. 
5. uXla P. e. BA^'] B anpra 
MHj ih Kttl r V. ArE] BUj^ra^r 
r. ieziv fojj] om. V. x«i ^ vito — 



rP. 



4 



I 



I 



I ELEMENTOEUM LIBEH VI. 85 

in triangulo BFE imi laterum EF paraJlela ducta. 
est j4A) [prop. IIJ, erit etiam SA : AF = BA : AR 
[V, 11]. quare AV = AE [V, 9], quare etiam 
iAEr= ArE[J, 5]. sed L AEr= BA^ esteriori 
p, 29], et i AFE = rA^ alterno [id.]. quare etiam 
L BAJ = FAJ. itaque L BAF recta A^ in duaa 
partes aequalea sectus est. 

Ergo Bi anguhis triangi^ in diMs porteB aeqaales 
diuiditur, et recta angulum aeea&B etiaiii baaim aecat, 
portes bssiB eandem rationem habebunt ac reliqua 
latera trianguli; et si partea baeis eandem rationem 
habent ac reliqua latera trianguli, recta a uertice ad 
punctum seetionis ducta angulum trianguli in duas 
partes aequales aecabit; quod erat demonstrandum. 

IV. 

In triangulia aequiangulis latera aequaies anguloa 
comprebendeutia proportionalia sunt et correapon- 
dentia, quae sub aequalibua angulis subtendunt. 

Sint trianguli aequianguli ABF, JFE habentes 
LABr= ^rE,BAr=r^E,ArB'=rEJ. dico, 

iazw f(rr)] om. B et Y (raa. eet quattae partia linea.e); in lag'. 
ttftnaaunt in ras. p. 10. ^] om.^ V. Sixa\ om. F. 11. 

njv -/avlav] Pj avrfiv BPVp. (^■frefa] mg. m. [ P. it- 

^ri P et »6q. raa. 1 litt V. 18. lo] m. 2 F. 13. xal 

Uv - 17: Srtiai-\ In ras. m. l_F. U. hvl ^orr. ex h^, p. 
loyov t%V ^- '■''■ ^O" XQiytavoti] om. FT. 17. yaitiav] 

tv9tiav p^. 20. at Jiej»^] e corr. V. ftas] m. rec. F. 21. 
nXivifai vnozsCvovaat Bp, vTtoxiivovaai jtXfVQoi FV. 22. 

fonnooi' V. zfTE] TJE Bp, V m, 2. 23. ^BT] 

EAF P. ytoviav'] comp. mg. P. ^FE'] TiJE P. 
B^r] BPp, V m. 2; BFA P; ATS V m. 1. TJE] 
T m. 2; rEJ P. -«rB] Bp, V in ras. m. 2} " " 
26. r£J] BFp; ^ET io ras. m. 2 V; ^VE P. 



21. ■ 

] Bfp. J 

srpF. ■ 



86 ETOiXEiaN s'. 

avdlojiov Eiaiv at nXEvgai ai jr^pt ras teag ytavCecg 
xal oftddoyot «f vjtb xaq Caas yavias vxottCvoveai,. 
Ksic&a yaff in svQtiag ^ BF ty PE. xal izsl 
at vnb ABF, AFB yaviai dvo opS-dv ^Aarrov^ 
j eCsiv, ifSri 61 ij vjib AFB rf; imb AEr^ ai < 
vjtb ABr, JEF Svo oQ&av iKatxovis eiaiv 
EjJ apa ixfiallonevai- avitnsOovvtai. ix^E^l^^a&atata 
Xttl avp,ni^TiT<aaav xara cd Z. 

Kal ixBl tenj iazlv ij vab AVE yavia rij 

10 ABT, Tttt^alXijlos ieriv ^ BZ r^ TA. ndXiv, ixA 
tai\ iarlv Tj imb APB TJj vab AEP, itaQalXrilos 
iariv ij AF tjj ZE. aapalX^loyifafifiav aga iarl 
rb ZAF^' tajj uQa i) jtiv ZA zfi AF, ^ 8e AF t^ 
ZA. xal intX rQiyotvov zov ZBE jrnpff (liav rijv 

16 ZE tjxrai ^ AF, iariv aga mg 17 BA wpog ti]v AZ, 
oSrms '>i Br JtQOs tjjv VE. tarj S} 1/ AZ ri} V^' 
ms op« V BA «pog t^v F^, ovrag i, BF jrpoj t^w 
FE, xal ivaXla^ ras ^ AB x^bg Wjw BV, oSrros ^ 
AF npbg riiv FE. JtaXiv, intl iraptf/AijAdg ioriv 
a i] FA TJj BZ, ICTtv aga ms ij BF itffbs tijv FEj 
ovtas i\ ZA jrpog ti^v ^E. taij 6i ij ZA TJJ AT' 
rag apor 71 BF wpos tijv FE, ovrojg ij AP itgbs r^v 
iJE, xal ivalla^ ros ij BF mpbg r^v FA, ovrtag 
i] FE arpo? i^v EA. izBi ovv iSsCjPri rog (ikv ij 

25 AB apbs ti]v Sr", ovtrog ij ^F npbs rijv FE, «Sg 
Sh ij BF Kpos rijv FA, ovtas v FE repos t^v E^m 
Si taov aQa as tj BA nrpog tijv AT, o^tas i\ TM 
itQos ti]v AE. 



10. itniv] 1 



. duol atdvoP, 



orr. m. 1. lldeaovss ' 

Sga laxiv BVp, P i 

c/] iariv P, comp. p. 
raa. p; FiJ V, corr. 



iL in 

rin 

14.' Z.if] 'i 



6. JluaaovEgT, 

SeqnQntia in 

t. ZAr^riB. 



ELEMENTOKUM LIBEK \T. 



87 




- iu trianguHa ABF, zlFE la- 
tera aequalea angulos com- 
preliendentia aequalia esse et 

correSpoiideiitia, quae sub 
aequalibus augulis BubteDdant, 
ponatur emm BF in pro- 
ducta PE, et quoniam 
[ l, ABF + AFB duobus rectis minores sunt [I, 17] 
f et L AFB = JEF, erunt L -^BF + AEF duobua 
f reetis minorea, itaque BA, EA productae concurrent 
[I atz. 5]. producantur et concurraut iu Z, 

et quoniam L AFE = ABF, erit BZ rectae 
r^ parallela [I, 28]. ruraua quoniam L AFB = ^Er, 
erit AF rectae ZE parallela [id,]. ZAFA igitur 
parallelogrammum est. quare ZA = ^T, AF = Zd 
[I, 34]. et quoniam in triangulo ZBE uai lateri ZE 
parallela dueta est AT, erit BA : AZ = BT;TE 
[prop.II]. sed^Z = r^, itaque S^:rz/ = Br: Tfi 
et permutaudo [V, U)] AB.BT^AT: TE. ruraus 
quoniam TA rectae BZ parallela est, erit BT-.TE 
"= Z^ : ^E [prop. 11]. sed ZA = AT. itaque 
&r:TE=AT: AE, et permutando [V, 16] BT: TA 

^■= TE : E/t. iam quoniam demonstratum est, esae 
AB : BT = AT: TE et BT : TA = TE:EA, ex 
aequo erit BA:AT=rA: ^E [V, 22]. 

^Z P. ZBE] PF, V m. 1; BZE Bp. V m. 2. ^fav 

T»» nZeupm» V. 16. ^] (alt.) om. P. tijiO ■om. BFp. 16. 
rjji'] om. BFp. 17. !«»•] om. BFp. i^i-] om. if. 18. 
.,*B] B^ p. «floe iiif] PV; n^oc BFp, et aio deinde 

per totam propOBitionem. 21. ZJ~\ (alt.) /JZ V m. I; corr, 
m. 2. 23. Kttl IvttXltti] P; JfBWaf upa Theon? (BFVp); 
ofr. lia. 18. 24. tml ovv\ xnl txsl P. 17 fi^f P- 27. 

Ndl *»' Piiow P, 



rroisEifiN s'. 



Tcov apa taoymviav Tqiymvav uvaXoyov siOtv xt 
alsvpttl al %sqI ta^ ttsaq yavias xal ofioloyot aC 
VTto Tiig fiJas ytoviag vaozsivoviSaf ojifp sdsi SBC^ai, 



'} 'Eav Svo XQlyavK zag JiXsv^aB dvaXoyoi 
iXV-' iOoydvia iOxai xii x^iymva xal COag ^gi 
ra; yaviag, i^' ag aC hnoKoyoi, aksv^X vnt 



1 



"Esxia Svo t^iymva xa ABF, /3EX xa^ xAfupibg 

10 aval.oyov s%ov%a, Ag (liv f^v A& apdg t^v BF, ov* 
Ttag z^v ^E npog Tiji' EZ, tog Si zijv Brnpog zijv 
r^, ovtag r^i/ EZ nQog T171' Z^, xal hi ag t^v 
RA TtQog ti\v AV, ovziag tijv EA tcqos trjv z/Z. 
lEya), ozi ieoytoviov iozt x6 ABF xpiytovov xa ^EZ 

16 tpiytova Jtal fffag e^ovei. zag yatviag, vtp ag at hjio- 
Xoyoi aXsvpal vnoxsivovaiv , t-^v iiiv vjm ABT z^ 
iiTtb AEZ, tiiv Si vnb BFA zy vab EZA xal StM 
Tijv vnb BAF t^ VTta EAZ. fl 

2^wsistcLtt3 yttQ aQog xy EZ Bv&eia xal toEs nif6gM 

20 aixf] etifisiois TOts E, Z i^ jik' vno ABF ytovttf 

Ce-q ij vxb ZEH, tjj ds into AFB iWij 17 vitb EZH' 

lomij apa ^ n^bg xa A loi.ny t^ npbg tp H ieztv fffij. 

teoytoviov aga iazl ro ABF zqiymvov xm EHZ 

[tgtymv^]. tmv apa ABF, EHZ tQiymvav avaXoyov 

25 tieiv at nksvQttl at xsqX xicg Ceag ytoviag xal 0^6- 



3. vno] itifC p. yavias] biB p. nXtVftiX vxotfCvovaui 
BFp, vnOTttvovaut jilingai V. 7, luj] m. rec. P. 10. 

117» BT] srBFp. 11. ij5» EZ] EZ BFp. i^* rj} 

TA BFp. la. ovnD B. u]» Za^ P, V m. 1; 117» aZ 

T m. S; JZ BFp, _ 13- ovzm Bp. i^* JZJ V; iij» ZiJ Pj 
JZ BFp. 14. fomi P, coiiip, p. 16, uKOie^^ouDi Vp. 



iri^ 



r 



I 



ELEMENTORUM UBER VI. 89 

Ergo iu triangulis aequiaiigulis latera aequales 
angulos comprehendeutia proportioaalia auut et corre- 
flpondentia, quae sub aequalibua anguliB subtendtmt^ 
quod erat demonBtranduin. 

V. 
Si duo trianguli latera proportionalia habent, 
aequiauguli erimt trianguli et eos angulos acquales 
habebunt, sub quibus correspoudentia latera sub- 
tendunt. 

Sint duo trianguli jiBF, ^EZ latera proportionalia 
^ babentea, ita ut sit AB : BF 

= ^E : EZ, Br i FA = EZ 
■.ZJ,&A:Ar=EJ:/IZ. dico, 
triangulos ABF, AEZ aequian- 
gulos fore et eos angulos aequa- 
les habituroa ease, sub quibus 
correapondentia latera subten«iant, 
i ABT= ^EZ, BFA = EZA, BAT^EAZ. 

eonBttuatur enim ad rectam EZ et puncta eius 
E, Z angulo ABF aequalis L ZEH et angulo AFB 
aequalia EZH [I, 23]. itaque qui relinquitur, anguiua 
ad A positus reliquo angulo ad H posito aequalia 
est [I, 32]. itaque ABF, EHZ trianguli aequianguli 
sunt. quare iu triangulis ABT, EHZ latera aequalea 
- anguloa comprehendentia proportionalia aunt et c 

21. AFB} e corr. V. 22. n^oe t£ A] P; ino B^T Tbeon 

(BPVp). npoe T<5 H] P; rwo EHZ Theon (Bp; uno EZ 

B BUpra ecx. H V, una EZH F). 23. ItsoymvK' F in fine lin. 

K iotir P, comp. p. EHZ] P, V m. 1; ZEH Bp, V m. 2, 

H y eras. Z et H. 24. iQiydva,] om. P. EHZ] P, V m. 1; 

^H ZEH BFp, Vm. 2. 




90 



STOIXEIiiN C. 




Xoyoi aC vno x&s iaag yavias VTioztCvovOm.' 
aga rog ^ j^B arpog tijv BT, [ovrrag] ^ HE arpos 
i^v £IZ. R^^' toff ij AB OTpog T^v Bf, OUTOJS lOTd- 
xaitat ^ ^fi ^p(';s r^i' fJZ' mg apa ij ^E apig 
5 tiji' EZ, ovTtas 71 HE jrpoj t^v £Z. sxatiqa aga 
Tiov jJE, HE 'JtQos tijv EZ xov avrov ix^t i.6yov; 
iOTl a^a istlv jj /ilE zij HE. Sia ra avxa Si] xal 
A2. r^ HZ iflriw fOij. i%il ovv fiHj iazlv )j ^B 
SH, xon'^ dJ ^ £2, tfijo «S^ af ^E,EZ Svt§l 

10 if£, EZ fffai etfftV xai ^aVi? ^ ^Z ^dosi Tp Zi 
[iffrtv] teij' ymvia uQa ^ uno ^EZ y<avia tij v: 
HEZ ietiv tat), xal t6 .JEZ tQiymvov xa HEZ' 
tQiyava isav, xal al XqixoI •ycovCai tatg JiotJtats 
yiDviais leai, vip' «g aC iSai alEVQal vxotsivovaiv. 

J5 tCi} aga iotl xal ij iilv v%o ^ZE yavia rij vno HZE, 
7] di vab E^Z t^ vxo EHZ. xal i%s\ tj ^ev v7t'o 
ZE/i ry vito HEZ iortv tOri, all' 17 wro HEZ t^ 
vno ABF, xal i} vxo ABT apa yavta r^ V7t6 ^EZ 
istiv i'07j. Sia ta avta Sij xal ij iab AFB ty vxb 

20 ^ZE iativ fffij, xal itt ^ ^Epog ta A tij srpoff rp 
^" iaoyiovLOv UQtt iatl tb ABF XQiymvov rp ^Ei 
tQtydva. 

'Eav apa Svo tpiyava tas aXFupag avdXoyov i^j 
iffoymvia sfftat ta tQiycava xal taas i^tt tag ycovias, 

26 vip' ds at bfioXoyot TtXsvgal vnoteivovatv' owfp ISct 
Setlai. 



xga 

i 



tQtya 



1, yaiviag] m. S 1 
2. T^v] om. BP|). 
ali' — 4; EZ] mg. 



(liav y 

nXevgal vjcozelvQvaai Tbeoi 

«.•«,] o». P. 3. .M 

. 1 P. 3. i^.] om. BFp. 



("« r 



3 

«s, 

1 



(BYFp). 

>iii. BFp. 

4. *jj*] 



ELEMENTORUM UBER VI. 91 

spondentia; quae sub aequalibus angulis subtendunt 
[prop. IV]. erit igitur AB : BF = HE : EZ. sed 
AB : BF = JE : EZ, ut supposuimus. quare 
^E:EZ = HE : EZ [V, 11]. itaque utraque ^E, HE 
ad EZ eandem rationem habet. ergo JE = HE 
[V, 9]. eadem de causa etiam JZ = HZ, iam 
quoniam /JE = EH^ et communis est EZ^ duae 
rectae JEy EZ duabus HEy EZ aequales sunt; et 
JZ = ZH. itaque L ^EZ = HEZ [I, 8], et A JEZ 
= A HEZf et reliqui anguli reliquis angulis aequales, 
sub quibus aequalia latera subtendunt [I, 4]. itaque 
L ^ZE = HZE, L EJZ = EHZ. et quoniam 
L ZEJ = HEZ, et L HEZ = ABT, erit etiam 
L ABr= jdEZ. eadem de causa erit etiam L AFB 
= jdZE, et praeterea angulus ad A positus angulo 
ad zl posito. itaque trianguli ABF, /iEZ aequi- 
anguli sunt. 

Ergo si duo trianguli latera proportionalia habent, 
aequianguli erunt trianguli et eos angulos aequales 
habebunt; sub quibus correspondentia latera subten- 
dunt; quod erat demonstrandum. 

VI. 

Si duo trianguli unum angulum uni angulo aequalem 



om. BFp. xal coff a^a P. 5. tiJv] bis om. BFp. 6. ifE] 
Eif V. 7. TCf] om. p. 8. teifi lexiv p. 10. disi Vp. 

JZ^ ZJ P. ZH] post ras. 1 litt. V. 11. lexivl om. P. 
13. Post Xqqv add. hiszi Bp, F m. 2, V m. 2. 14. Post f<yai 
add. Uovxai Bp, F m. 2. 16. laxiv PB. z^ZE] ^EZ F. 

ifZE] H supra m. 1 F. 17. tiiy\ icxiv ©. aUa P. 18. 
ABF] (prius) ABF hxiv tffrj V. 19. 17] 17 fiiv P. AFB] 
ABF ^. 20. §xi] e corr. V. xm] bis x6 B et V (corr. 
m. 2). 21. J iaxiv tarj FV. iativF. 



92 

via taijv 



rroiSEiiiN ?'. 



I 



7ii.ivpag avttloyov, COoyavta tatat xa Tp*'- 
yava xal fOag £|ce rag yaviag, v<p ag al op6- 
Aoyot xXsvpal vvotBivovGi.v. 
'j "Etsxa &V0 rpiyutva ta ABF. AEX fiutv yatviav 
t^ vno Bjir (UR yavCa rjy vxo E/3Z ftfijf Ixovxa, 
iltQl 6i rag Isuq yarviaq ras altvpag ava)ioyov, ag 
t^v BA nqoq t^v AF, oikas tijv E^ npos Tijv jJZ' 
itya, on laoydviov ieti zo ABV Tgiyavov r^ A. 

10 xpiymvixi *ai iaijv £|ct i^v vxo ABF yaviav 
vito ^EZ, ri]v 61 imi) AFB tjj vnit ^ZE. 

I^vviardro} yccQ itgos r^ AZ tv&£ia xal Tots »( 
avr^ 6ii(ttiois rorg ^, Z oxorepa fiiv rtav vno BAT, 
EAZ ISti ij i-xb ZAH, rfj di vxo AFB fgf) ^ vnb 

16 ^ZH' lomi} aQK i] spbg tm B yavla Xoix^ r^ 
wpog tco H tei) ivriv. 

'laoymvtov apa isrl rb ABF xffiymvov rm AHZ 
tgtymvp. avakoyov agtt iarlv tog tj BA n^o^ rijv 
AF, ovxtog ^ H^ ngiig rijv AZ. vaoxiitai 8i xal 

20 wg ij BA aQog rijv AT, otrrcffg ij EA Xffbg rijw ^JZ' 
y.al ag uqu ij EA itQog xijv dZ, ovrtag ij H^ JtQbg 
rijv AZ. Carj aga ^ EA rfj AH xal xotvij ij AZ' 
dvo dii ul EA, AZ 8vaX xalg HA, AZ feat tlaiv 
x«l y&via ii vab EAZ ytavit^ r^ vxb HAZ \iaxtv\ 
5 iejj' ^datg Squ ii EZ ^aati, rft HZ iertv Cai], xul 
rb AEZ XQiyavov xca HAZ tQtymvip taov ieriv. 



7. fffBs] m. 2 7. 8. xnv ^r] Ar BFp. jteos] aad 
m. ree. P. cijr] om. BFp. z/Z] era«. V; mutat in jDS^ 
Z^ Bp. fl. lativ P, comp. p. 10, tmv ABT F. 

t^i-] TB V, corr. m. ree. A TB] e corr. m. 2 ¥. l^ — 
jiiv BFVp. riiv i)Z tv&tiBv V, corr. m. 3. ts, avr^s3 




f ELEiMENTOEUM LIBER VI. 93 

habent et latera aequalea angulos coniprehendentia 
proportionalia, aequianguli erunt trianguli et eos 
augulos aequales habebunt, sub quibus correspon- 
, dentia latera Bubtendunt. 

Sint duo trianguli ASF, ^EZ unum angulum 
BAF uni angulo E^Z aequalem 
habentes et latera aequales angulos 
^n- comprehendentia proportionalia, ita 
ut sit Bv4:vVr= E^:z/Z. dico, 
triangulos ABT, ^}EZ aequian- 
guloa esse et habituros esse l_ AB F 
= ^EZ, LArB = ^ZE. 
construatur enim ad rectam ^Z et puncta eius 
rf, Z utrique angulo BAF, E^Z a«qualia i Z^H 
' et L AZH = ATB [I, 23]. itaque qui relinquitur 
angulus ad B positus reliquo angulo ad H posito 
aequalis est [I, 32J. itaque trianguli ABF, /iHZ 
aequianguli sunt. quare erit BA : AT = H^ : /IZ 
[prop. IVj. supposuimus autem, esae etiam BA : AT 
=- E/f : 4iZ. quare [V, 11] EA : /tZ = H^: z/Z. 
itaque £^ = z/H[V, 9]; et communia eet AZ. itaque 
duae rectae'£z/, z/Z duabus HA, AZ aequalea aunt; 
et L -E^Z = HAZ. quare _EZ = HZ et A AEZ 
-= H^Z, et reliqui anguli rehquis aequales erunt, 

1*. E iJZ yavia % V, 16. x6] zo V, corr. m. 3. yaviic] po»t 
raa. 1 litt. P; om. Theon (BPVp). 16. i^] x6 V. corr. m. 2. 

17. tativ Pcp, comp. p. JHZ] JEZ q>. 18. tM om. 
BFp. 19. HJ] litt. .H m. 2 V; EJ B. corr. m. 2. xvv] om. 
BFp. 20. t^*] bts om. BFp, EJJ JE F; HJ B, corr. 
m. 3. 21. EJ] BJ <p. x^v] om. BFp. JZ] ZJ V, 
cort. in. 2. HJ] ex *JH m. rec. P. 82. x^v] om. BPp. 

fl3. elai Vp. 24. yavla afa F. ianv] om. F. 25. 

MZ] ZJI P- 2G. ioii BV, comp, p. 




94 ETOISEIHN 5'. 

al Xomal ymvCai rofs Xomais •yetvlaiq toat BOovrm, 
vtp' ag aC l'aai xlEvgal vmot£Cvovei.v. te^ «pa isrlv 
^ fthv ^ffo ^ZH tTi ijjco ^ZE, ^ Sl vao ^HZ t^ 
vjtb /JEZ. aXl' r, vjto zfZH t^ vao AFR iexiv 
B fffT)" xal i] vno ATB «p« tfi wi6 AZE ietiv leij. 
vnoxiitat di x«l ^ vjto BAV r^ vxo EAZ torf xctl 
XoLJtrj apa 7] %QOS ta B Aoin^ rg JcpOg ta E teij 
ietCv looyioviov «p« ierl to ABF tpiymvov ra /3EZ 
tifiyatvst. 
10 Eav «pa Svo tptywva fttav yavtav (ua yavi 

teijv ixtl^ ^^9^ ^^ ^'^S teag ywviag tag nXsvQag av^'< 
Xoyov, ieoymvia ietai ta tgtymva xal teag t^i 
ymviag, vfp' Sg «f 6fi6Xoyot xXsvpal VTCOttlvovai 

» c 

'Eav Svo TifCyava fiiav yaviav fita yaviu 
taijv ^xVf ^^Q^ ^^ aXXag yaviae t«$ xXavpag 
«vaXoyov, tav Si Xontav exatigav a fia ^rot 
iXaeeova r fiii iXdaeova oQ&^g, ieoymvta 

20 ietat ta tpiyava xal leag ^|«t tag ymvia\ 
JTfipi ag avaXoyov slaiv ai' nXev^ai. 

"Eeta Svo TQiymva tk ABF, AEZ (iCav 
viav (itu yavia taifv ixovta triv vito BAF t^ ixi 
EAZ, fffpl dh aXXag ymviag rag vxo ABF, jdEZ 

26 zag JcXevQag avaXoyov , ag ti\v AB XQog t^v Br, 
otJTOg. ti\v AE TiQog tijv EZ, trow Si XotJtav ti 



1 



dEZ 
BF, 



1. teitvten lnaTeQa exttr^pa Tbeon (BFYp). _, ...^ 

^HZ] Peyrardns, imo dEZ P; Ttifit ttS H Theon (BFVpi 
To pro Tio V, corr, m. 2). 4, vno ^JEZ] PejrardnB; vxi 

^HZ Pi npDs T(5 E Theon (BFVp; to pto ira V, coir. m. 2). 
aXla P, ATB] BVA V, A in ras, 6. xol ^ — lirciv finj] 



■ ELEMENTOETJM UmER VT. 95 

aub quibus aequalia latera subtendunt [I, 4]. itaque 
L ^ZH = /IZE, L ^HZ = ^EZ. uerum L ^ZH 
= AFB. quare etiani L AFB = dZE. aupposuimua 
autem, esse etiam L BAT = E-dZ. itaque etiam qui 
relinquitur angulus ad B positua, reliquo angulo ad 
E poaito aequalia est [I, 32]. itaque triaoguli ABT, 
jdEZ aequianguli sunt. 

Srgo si duo trianguH unum anguluni uni angulo 

aequalem babent et latera aequales anguloa com- 

prebendeutia proportionalia, aequianguli erunt trian- 

guli et eos angulos aequalea habebunt, sob quibus 

-Teorrespondentia latera subtendunt; quod erat demon- 

kndum. 

vn. 

Si duo trianguli unum angulum uni angulo aequa- 

lem habent et latera alios duos anguloa comprehen- 

dentia proportionalia et reliquoa augulos singulos 

simul aut minores aut non minores recto, trianguli 

aequianguli erunt et eos auguloa aequales habebunt, 

qoos latera proportionalia comprehendimt. 

^^ ^ jf Sint duo trianguli ABT, AEZ 

^B A A unum angulum uni angulo aequalem 

B I \ -J \ ^i*'^^'!*^^^'^-^^^^-^'^^, etlatera 

^V / ^'^^'"Ov "■'^''^ duos anguloa comprehendentia 

^tt____\ ^2 proportionalia,^5 : BT-=^E:EZ, 

^V ^^^^^ et reliquos angulos, qui ad T, Z 

Hl J' positi sunt, priua aingulos simul recto 

, «tD. p. 7. r^] rd P. xm) e corr. P. 8. ItnQ Itniv P, 

'oomp. p. 19. IXdTzova b\B F. FriuB llauaova cort. ei 

fXaaaov m. 2 P. 23. fttn ycoviR] punotis notat. F. 24. 

£J2] oorr. ex ^BZ m. rec. P. ' jiBr] BAF ipi ^SJ p. 

85. xiiv BF] Er BKp. 2e. t^v EZ] KZ BFp. 



ETOIXEIiiN S", 



1 

! Op- 

V xp 
AouriH 

jto 

1 



;T£i6g Torg r, Z afiotEQov ixtttdpav «iia iXaaaova 6q- 
&iis' ^iyOf 0T( iaoytoviov iezi to ^BI" TQfymvov tp 
^EZ tpiytova, xal Larq larai ^ vab ABF yavia x^ 
vxb /iEZ, xal Xomij dtjlovott. ij mpoj ta F i,oi 

5 t^ ZQOs tto Z fOi;. 

£^ yap aVLaog iettv ^ WJto ABF ytovU 
AEZ, ^Ca avrav ^iti^av iatCv. Sata> fif/£(ov ^ ijto 
ABV. xal aweatdta agbg tfj AB tv&&ia xal 
ngbg ttvtii aTjfiBi^ tto B ty {nb AEZ yavla Iot] 

10 vxb ABH. 

Kttl intl Hstj ietlv ri {liv A ytavia ti] A. 
v%b ABH tij vnb tdEZ, }.oini\ «pa ^ trjro AHi 
loiJtij tfj vnb dZE iativ t07]. laoytaviov «Qa isnl 
to ABH tQiyavov tra ^EZ tQiytavoi. Sottv Kpa cos 

15 71 AB iiQhs tiiv BH, ovtfag t] .dE XQog tijv EZ. 
cog di ij ^E JtQoq tijv EZ, [ovtag] wtoxeittti i] AB 
jrpos triv BV" 7] AB «pa «pos ExatiQav naii BF, 
BH tbv ttvtbv i'x^i. koyov tajj apa ij BF rj; BH. 
aatd xa,l yapia r] jtpog ta F yavi^ t^ vnb BHF 

20 iativ tarj. ikattav &h op&^g vntoxettat ij xpbg tto 
T" ii.ixtrav «pa iatlv opS-^s xal i] VTib B HF' aatt 
i} ifpslijg avtji ycavia i] vjtb AHB (isiitav iatlv 
dpftijg. Kttl iStlx^T] fay] ovaa ty ngbs ta Z- xttl 
7] icpbg t(a Z uQa ^ei^av iatlv opO*^?. vitoxsitai. 

25 ds ilaaaav OQ&ijg' ontQ iazlv atoreov. ovx UQa 
aviaog iatLV i] vab ABF yavCa tj7 vab ^EZ' tat) 



S. ietiv P, comp. p. 3. ictai] iazlv ^^| 



10. ARH'^ H 8 eorr. p. 13. yravia ts V. 13. loi. 

aupra m, 1 F. iiSTi] comp. p; lot/v PF. 15. t^v] . 

om. Bf p, 16^. ^s m inoKsivai 6i xoi v>s Bp. x^] 

oni. BFp. ovitoi vxo>senui] iiiotientn FV; oSims Bpi 

"i03C P. 17. i^v] om. BPp. Post BP ftdd. 



^^ 



^m ELEIJENTORUU LIBER VI. 97 H 

minores. dico, aequiangulos esse triaDgulos ABF, V 
^EZ, et L^Br= ^EZ, et, ut inde adparet, qui I 

relinquitur aagulus ad F poaitua, reliquo angulo ad Z I 

Iposito aequalem esse. H 

nam si i .-iBranguIo ^EZinaequalis est, alteruter H 

eorum maior est sit maior i jiBr, et construatur 1 

ad rectam AB et punctum eiua B i ABH — ^EZ 
p, 23]. et quoniam L ^ = L ^ et L ABH = JEZ, 
erit LAHB= JZE [I, 32]. itaque trianguli ABH, 
/iEZ aequianguli sunt. quare AB : BH = AE : EZ M 

[prop.lV]. sed suppoauimua, esse ztE:EZ = AB:Br. I 
itaque AB ad utramque BT, BH eandem rationem H 
habet [V, 11]. quare Br= BH [V, 9]. itaque etiam I 
engulus ad F positus angulo BHF aequalis est [I, 5]. V 
snpposuimus autem, angulum ad T positum minorem 
esse recto; quare etiam L BHF minor est recto. 
itaque angulua deineepa poaitus AHB maior esb 

Irecto [I, 13]. et demonstratum est, eum angulo ad 
Z posito aequalem esse. quare etiam angulus ad Z 
positus maior est recto. supposuimus autem, eum 
recto minorem esse; quod absurdmu eat. itaque 
L ABF augulo ^EZ inaequalis non est; aequaiis 
igitur. uerum etiam angulus ad A poaitus angulo ad ^ 

Iposito aequalis est. quare etiam qui relinquitur angulus 
ad r poeitus, reliquo angulo ad Z posito aequalis eat 
P, 32]. ergo trianguli ABF, ^J EZ aequianguli aunt 
Theoa: «ai tJe ai/a tj ^B ngbs t^v BF, ovtois >i AB tcfie 
Tiif BH (V et bis omisBo rrv BFp). 18. affa ia-civ P. 

19. jtpos n5 r]_corr. es vno BHF ja. 2Y. BUF] corr. 
ei BFH m. 2 V. 20, Haatniiv p. 21. xo^ oni. f. 

22. aw^s P. 23. ico] corr. ex id m. 1 B. 26. lldi- 

lart- F. lativ] om. 7.' 26. -JEZ] BJZ p. 




98 ZTOrXEISN C 

iiga. iatt Se xal 17 irpos tj5 j4 lCij tjj wpoff i 
xal loiTiij apa t/ apog toj F Aoireij t^ Jrpog tS Z 
iai} iativ. iGoydviov apa iszl t6 ABT rQCyavov tja 
^EZ tpiymvo). 
5 WAAa d^ itdiiv vTCOKtiod-a ixatigtx tmv wpog rois 
r', Z f(^ iXdaamv oQ&iig' Xdya xdXiv, oti xal ovttaB 
ietlv iaoydviov t6 ABF ZQiyavov rp /JEZ tQiytova. 
TiDV ya^ avtav KUTaGxEvaa&ivtejv oftoitoff dsi- 
^oniv, ort Teij iozlv rj BF ttj BH' aCzB xal yavCa 

10 ^ jrpog ta r VQ vxo BHF 1'3jj idtiv. ovx iXdtttav 
3e OQ&ijg ^ Wpos ta F' ovx iKdtztov aQo. opd^^s 
ovtfi Ti v%o BHP. zQtydvov 6i} zov BHT aC Svo 
yoviai 6vo og&mv oijx slat.v iXdttoveg' OTtEQ ierlv 
dSvvazov, ovx ap« %d?.t.v avieog iattv tj vsro ABT 

16 ymvia zy vitb ^EZ- i^aij aga. Sati di xal ^ jrpos 
rp ji tfi Jtffog xm ^ /'ffij" koi,ni\ 'dqa i\ apog t^ V 
Xoiitri Tjj TtQoq t^ Z asij iativ. iaoymviov aga istl 
t6 ABF t^iyavov Tp ^EZ tQiymva». 

'Edv «pn Svo tpiyava ^Uav yavlav (ttie ycavt^ 

20 fojjv fj;g, Kfpt di dlXas yaviag tag xlsvpag dvdloyov^ 
tmv 8i Xoiittov ixaxiQav «(la ikdtzova ^ fnj ikdtttivfi 
opfr^S, taoytavttt itfrat, zd z^iyfova xal laag e^si zas 
yoviag, asgl Hg dvdXoyov bIgiv at nlevQai' oas4f 
SSsi Saiiai. 



'Ettv iv OQ&oymvio} TQtycovoi dxo rijg o^ 

fHjg ycaviag iitl tijv ^aatv xd&stog aX'9fl, ta 

1. laiiv B. Po3t A ad(i. ejnLtCia Bp, eupra F, i 
3. hzil iarlv P, comp. p. 6. flacttov F. 710, 

TCLt V. 7. Caoymviov imiv P. 8. ouoCms i^i) BVp. 
fiaooiD» p. 11. HdvUBv p. 12, ovdi] om. V. ^] 



1 

S 

I 



] 




I ELEMENTOEUM LIBER VI. 99 

iam rurans supponamus, ntrumque angnlum ad 

J", Z positum recto niiuoreni iion esee. dico rursiiB, 

sic quoqne trianguloe ABF, /IE2. aequiangulos esse. 

iiedem cotnparatis aimititer demonatrabimus, 

esse &r ^ BH. quare etiam angulus ad F positus 

, aiigulo BHF aeqiialis est [I, 5]. 

angulus autem ad JT positua recto 

mioor non est quare ue i BHF 

_^2 qnidem recto minor est. itaque 

trianguli BHF duo anguli duobua 

rectia minores non annt; quod fieri 

non poteat [I, 17]. rursns igitar 

i ^jSrangulo ^EZ inaequalia non 

est; aequalis igitur. uerum etiam angulus ad A posi- 

tus angulo ad ^ posito oequaliB eat. itaque qui 

relinquitur augalus ad F positus, reliquo angulo ad 

Z posito aequalis est [I, 32]. ergo trianguli ABP, 

jJEZ aequiauguli suut. 

Ergo si duo trianguli unum augulum uni angulo 
aequalem habent et latera alios duos anguloB com- 
prebendentia proportionalia et reliquoa aagulos singulos 
aimul aut minores aut nou minores recto, trianguli 
aequianguli erunt et oos angulos aequalea habebunt, 

Iqnos latera proportionalta comprebendunt; quod erat 
demonatrandum. 
vin. 
Si in triangnlo reetangulo ab angulo recto ad 
> 
eo 
00 



1 



I 



I 



I S P. a^] 8i V. 13. ilaneovts V. 15. inlv PB; 

[ comp. p. la. fttj] bBCTt. postea F. 17. 1«/] UxCv PF; 
I oomp. p. 20. fj^] corr. «s i^jet m, 2 P. tks] om. T. 

■ 21. ana ^xm V. 26. aio] «nd V; corr. m. 2. 



J 



100 STOIXEIflN ff'. 






ngijg t^ xad^drp T^iymva Sfiota iat. 
olp xal alltjlois. 

"Eetca zQiymvov op&aydvLOv to ABV 
tTjv vno Bj^r ytov(aVf xal ijx&a uno tov A ial 
5 trjv BF xd&itos fj Ad' Xeym, oti ofioiov iotiv 
ixdtEpov tav AB^, Anir tgiytovaiv oXa rp ABT 
xal hi. aiA^Aois. 

'ETtEi yag teri iatlv 17 vno BAF ty vxo A/4B' 
6g9i yag ixattf/a- xal xotvij tav dvo tQiysovav xov 

10 ta ABT xal toij AB^ ij Ttgog ta B, Xomij Spa 
7] vTto AFB loi%\i tri vno BAd ietiv tari' luoytoviov 
aqa iotl to ABF t^iyavov t^ AB^l tQiysav^, 
sativ apa lag 17 BF vjtoteivovau tijv op&^v 
ABf ttfiyavov ngbs ttjv BA vnotsivovOav ti]v 

15 &iiv tov ABzJ tQiytovov, ovtag avti] i} AB v«o- 
rslvovaa tr^v nQog ta> F ymviav rou ABF XQLytA- 
vov ngig tijv B/i vnotsCvovaav T^f fffijv t^v vno 
BA/J ToiJ AB^ tQiymvov, xal hi i] AF Jipoj trp> 
AA vTtotiivQveav tijv nQog ta B yaviav xot,vif»\ 

20 tav SvQ tQiytoviav. to ABP «pa rp^ymvoi' Tp AB 
TQiyaiva (eoytaviov ti iati xal tag atQl tag tSt 
ytovias nlBVQag avaXoyov ix^'- oyioiov apa [ietV^l 
To ASr t^iyiovov tip AB^ tQiycav^. bfioitog 
^Etlo^fv, OTt xal Tco AAF tQiytovm ofiotov ieri ti 



•9- m 
o{! J 



1. imiv F. 4. vmviav] om. p. 6, BF] AP V. „_ 
J^ P. itxt FV. 8. vKo] poatea inB. F. BAV ytavla Pl 

AJB] AB^ V, con-. m. 2. 12. zm] corr. ei tio - ■ * 
ABj1\ B aupra m, 1 F, 13. BF] FB B et seq. raa. 

Tjji^J poBt raB. 1 litt, V. 14. ABF] P in ras. 

BA] m ras. m. 2 V. vnoteivovaav] oorr. ex iin 

m. rec. P; in ras. m. S Y. 16. inottivovtiat F, t otoKm 
17, BJ] BJ r^v F. imQxiivovattv zr\v firiji' t^ wpoe t^ n3 



■ ELEMENTORUM LIBER VI. 101 

basmi perpendicularis dueitur, triauguli ad per- 
pendicularem positi similes erunt et toti et in- 

8it triangulus rectangulus j^BF reetum habens 
igulum BAF, et ab .^ ad BT perpendicularis ducatur 
dico, utnimque triauguliun AB^, A/tF et toti 
BT et inter se similes esse. 
nam quoniam t BAT •= AiHB (uterque enim 
rectuB eat), et duorum trian- 
gulorum ABr, ABd com- 
munis est augulua ad B po- 
aitus, erit LArB<=BAJ 
[I, 32]. itaque trianguli 
ABT, ABd aequianguli 
sunt. erit '\^iwx BT\ BA = AB:B^ = AT: A^ 
[prop. IV]; nam jBJ^sub recto angulo trianguli v^Br" 
subtendit vi\. BA sub recto angulo trianguli AB/i, 
AB in triangulo ABT sub angulo ad T 
isito subtendit eiB^J ia triangulo AB.A sub angulo 
aequali BA^J, et AT, AA sub angulo ad B posito 
utriusque trianguli communi aubtendunt. itaqne tjian- 
guli ABT, ABjJ et aequianguli sunt et latera aequalea 
angulog comprehendentia proportionalia habent. itaque 
ABT r^ ABjd fdef. 1]. similiter demonetrabimus, 




sut 



Han 



.„. ^. - ,'. roiji' uBi^e F. 18./5J] viSrP. 17] inter 
duaB rai. F. PoBt AT add. F: imoxiliiovau xif» Rpog xa £ 
yat«^av tov AhT Teiytovav, eed del. id. 1. 19. vnoTeivoveai 

(t in ras.) poat ras. 1 litt. F, vxoTfivovta Bp, E] eeq, raB. 

1 litt 1. 20. uvTmv xmv V. Sfa] poatea ma. b'; m. 8 V, 

ABA Sfa V. ai. intiv P, oomp, p. 22. /orr| om. P. 

34. iativ F; comp. p. 




k 



102 ETOIXBIiiN S'. 

j4Sr xqCyavov ixdteffov aga zav AR^, A 
[tpiycjncw] ZyiOiov iaziv oXm xa ARV. 

Aiya 5ij, oxi xkI «A^^^otg Effrlv oiioicc xu AB^, 
A^r TQiytovK. 
5 'Eaifi yag opS^ ^ vao BA\A opO^ t^j ureo AjdF 
iaxiv i'6-ii, aXXcc fiijv xal ij t>«b BAA x^ npog tp F 
lSEi%&ri tST], xal Xomi} aga ^ repos rra B /otJtj; t;J 
vKo ^AV itftiv igr}' iooycoviov aga iari xb ABA 
tgiyavov xa> A^F tQiydva. iativ aga ag ^ BA 

10 roi; ABA tQiydvov vnoteivovGa rijv vjib BAA Ttgog 
xriv ^A zov AAV tQiyatvov vaotEtvovGav tjji/ ttQog 
ta r (611V tij vnb BAA, ovxats avtii ^ A^ Toi3 
ABni tQiydvov vxot€ivov6a xi}v repog tjo B ya- 
viav JtQog tijv AV vnatsivovaav tiiv vxo AAP xov 

15 AAV xQiytavov taijv T^ jrpog ta B, xal ht ^ B. 
^Qog rijv AF vnoTiivovtSai xag op&dg' o^ioioi' ai 
iatl tb ABA xgiyavov tc5 AAF tQiyd: 

'Ekv apa iv 6$9oyavia tgiydvo} anb tijg op*^ 
yaviag in,l tijw ^deiv xd&etog «x^j/, ta agbg tfj 

'<•> xa&itKi tglyava o^oid ian t^ te dA^ xal a^AijAoig 
[oTteQ Mei dd^at]. 

IIoQtOna. 

'EJK di) rovrou tpavsgov, oti iav iv OQ&oyoiv^ 
tQiyav(p d.Ttb t^g oq&^s yoivias ijtl xtjv pdoiv xd&ai 
25 log «X'^) '/ ttx^BtOtt tdv T^s ^desag t[i7i(uiTaii 
(ligfj dvdloyov iettv onsg iSst Set%at [xaX iti ■ 

]. T^^yiDVDv] oWl. BFp. 3. zfiiyrovotv] om. P. ofisii 
iaiiv Sim] om. V, ABF ■tQtydvta altp Sfiotov lattv '^ 

6. BJ'a] B e corr. m. 2 V. l'. loiny] corr. «t la -' 
. 1 F. 8. hiC] icT(v PF. 11. i^* ^A] z^ aA F; 



^(4 



I 



r ELEMENTOEIII LIBEK VI. 103 

«886 etiam t^AAT'^ AET. ergo uterque triangulus 
AB^, AJT triangulo toti ABT Himilis est. 

iam dico, trianguloa AB^, A^T etiam inter se 
^imiles esse. 

nam quoniam L BAA = A^T (recti enim), et 
demonstratum est, L BAA angulo ad T posito aequa- 
lem ease, etiam qui reliuquitur angalus ad B positus, 
angulo ^AT aequalia erit [I, 32]. itaque trianguli 
ABA,AAT aequianguli sunt. est igitur BA : AA 
— AA : AT ^ BA i AT [prop. rVjj nam B^ in 
triangulo ^Bz/ aub BA^ eubtendit et AA in trian- 
gulo AAT aub angulo ad T posito subtendit angulo 
BAA aequali, et AA in triangulo ABA sub angulo 
ad B posito aubtendit, AT autem in triangulo AAT 
8ub j^AT angulo ad B poaito aequali, et prae- 
terea BA,AT sub rectds angulis subtendunt. itaque 
AABAr^AATiA^i. 1]. 

£rgo si in triangulo rectangulo ab angulo recto 
ad basiin perpendicularia ducitur, trianguli ad per- 

Ipendicularem poaiti similea erunt et toti et inter se. 
Corollarium. 
' Hinc manifeatum est, si in triangulo rectangulo 
b1> angulo recto ad basim perpendicularis ducatur, 

m, rec. 14. vitotcCvovaciv\ -v eras. F. 16. t^] corr. ex 
T^S m. reo. P; Beq. raB, 1 Utt, V. 16. n^oi tl)f ^T] in 

I&B, P. vnoxelvovaa F. aO. iariv V. 23. iv] Om. p. 

26. TfHjfiaro)»] 0111. p. 26. tati B, oomp. p. 3xf( I6tt 
hrlStt»] om. BFp. «ai hi — p. 104, 2: ioi.*] postea ins. m. 
"T in raa; mg. m. 2 V. 



I 



104 ETOIXEiaN 5'. 

Paeemg xal ivbg onoiovovv tiSv Tfi^^fidreiv n : 
Tp T^-qfiati itXevQct fi^erj avdioyov ieziv]. 

&'. 

Trje do&{i6t]g tv&siag ro npaetax^^'^ /ispog 
E aipslciv. 

"Eezoi i] So&iiaa evQ-sia ^ ^B' 3st 3ri t^ff JS 
to 7tQO0tax&iv fiEgog ag>sXstv. 

'Emtstdx&ca di] ro tpirov. [««!] Si^x^^ ^'? (is»" 
Tou A sv&eta ^ AF -ycaviav wptej^ouffa (lera t^g 
10 ^B Tuxoi^flffV xal siki^ip&co tvxov ifrifistov inl z^q 
AF to J, ital xsicd-aSav zij AJ Itsai at ^E, ET. 
ml iaEt^x9ca rj BF, xal Sia tow ^ aaQcilkrikm 
avzij rix9(o ij ^Z. 

'Ensl ovv zQiytovov totJ ABT mapa ^iav taf 

15 ici.svQ<av T^v BF rjxtai ^ ZA, dvdXoyov aqa iotH 

mg ^ FA JtQog zyjv ^A, ovzatg rj BZ npbg t^v 

SiTtkii Sl ii r^ T^s A A' Si7tXi\ aQa xal ^ BZ i 

ZA- tptwA^ «pa rj BA rijg AZ. 

Ti\g aQa So&siarjg sv&siag zijs -^B to ^irtTcc^d'» 
20 TptTOv (tf'pog a^riQrjzai To AZ' ortiQ iSst Tloi^aa 

Trjv So&staav sv&stav arfirjzov rij 6o9e£il^ 
rstfiri[iivr} o^oCag zsfistv, 

X. Simplicins in phys. fol. 114', 116. 

I. oitotEgovovv F. S. Foat ^ifiii' eeq. oxfg (3ei del%tS 
BFp, V m. 2. 8. tpito*] ant« -rov ras. 2 litt. F. x 
om. P. Tie tv^cia btto tov J ij V. 11. tiiia^aiear'] !•»_ 
m. rec. F. 14. Snpra mtgd in P scr. m. rec. ^aedllrilSSt 

16. T^v\ t^ p. Zi/] mutat. in i/Z m. 2 V; zfZ Bp. IC 
t^* iJ^] Tjl ^^ B, JA Fp. iTjvJ om. BFp. 17. r^) 
T») p. xnl ij BZ T^s Z^4' rpinl^ npn] mg. m, I P. 

B.^] J in ras. P. 19. i^g] t^ p. t^s] corr. ei i^ 




ELEMENTORUM LBBER VI. 105 

ductam rectam mediam inter partes basis proportio- 
nalem fore. — quod erat demoil^trandum.^) 

IX. 
A data recta linea partem quamuis datam abscindere. 
Sit data recta j4B. oportet igitur 9,h j4B quamuis 
datam partem abscindere. 

sit data pars tertia^ et ducatur a puncto A recta 

jir cum AB quemlibet angu- 
lum comprebendenS; et suma- 
tur in -^r quoduis punctum ^, 
et ponatur ^E — A^ = EF, 
^ Z "S et ducatur BF, et per ^ rec- 

tae BF parallela ducatur ^Z [I, 31]. 

iam quoniam in triangulo ABF uni laterum BF 
parallela ducta est Z^, erit [prop. 11] 

r^ : ^ji = BZ:Zj4. sed F^ = 2 ^j4. quare 
etiam BZ = 2 ZA. itaque BA = 3 AZ. 

Ergo a data recta AB tertia pars AZ abscisa 
est, ut iussi eramus; quod opbrtebat fieri. 

X. 

Datam rectam lineam non sectam datae sectae 
congruenter secare. 

1) Nam demonstrauimns p. 102, 9 sq. BJ: JA ^ Ad 
: dV. reliqna pars corollarii p. 102, 26 sq. sine dnbio inter- 
polata est; nam et post sollemnem illnm finem demonstratio- 
num corollariommque oTrsp l^ft ^filat p. 102^ 26 additnr et a 
bonis codd. Theoninis aberat nec nsqnam nsni est. habet tamen 
Gampanns et P, qnamqnam sine clansnla illa. itaqne et in 
nonnnllis codd. ante Theonem et in quibnsdam Theoninis 
simul sponte interpolata est. 

20. rp^roy] in ras. F. 22. ^o^^icxil P, Simplicius, Gam« 
panus; ^o^hier^ hv^hia Theon (BFVp). 



106 rxoixEmN ?'. ^^^H 

'Eksrta ii [lev So&tt6a tv?ftla aTftJjrog ^ AB, 4i 
da TiTiiti^ivt] 15 AF xara xk jJ, E 07i(isia, xal 
XEie&aeav aOTB yavCav rvxovaav ne^iixeiv, xal 
imi,Evx&a ij PB, xal di,a rav ^, E ty BT nagal- 
& XtjXoi ^^td^offftv ttC z/Z, EH, Sia 6h roi» ^ tij AB 
naQallTjlog ^x^w rj A&K, 

IIaQaklr]X6yQa(i(iov apa isrlv ixazeQov %av ZSj 
&B- i'Gt] ttffa f) (ilv ^® Tj ZH, 71 Si ®K t^ Hi 
xttl ixtl tQiymvov zov AKV naQu (liav rav itKsv- 

10 ppv T17V Kr tv&iia TjXtttL jj &E, avaloyov Spa 
iexlv tog fj FE nffbg tijv EA, ourras ^ K& ^rpog 
XTiv &A. fffij Sl ti /ilv K& ty BH, ^ dh ®A ty 
HZ. lextv RQtt los 7j FE tcqos rrjv EA, ovxas ij 
BH ngbg xiiv HZ. ntiltv, inti xQtydvov xov AHE 

15 XttQa fiittv twv nXsvQav xrjv HE ijxxat jj Z^, ava- 
Xoyov aQa ietlv ag i] EA apog tijv AA, ovxias ^ 
HZ jrpog xijv ZA. iSiix^V ^^ ^*^^ ^S V ^^ Jrpog 
xi]v EA, ourtog i] BH xqos ti]v HZ' leriv «pa atg 
(liv 55 FE JtQog zi]v BA, ovtag ij BH nQog xijv 

20 HZ, tbs Ss 7] EA nQog rijv AA, ovTtog i] HZ ngog 
xrjv ZA. 

'H aQtt do&ftaa evQfLU ar[i7jrog ^ AB xs ^' 
&iiai] BV&ua rsxfiiKiivij r^ AP ofioici; ritff^Ti 
ojtEQ idii KOt^Cwf 



.irfiJo So&iteav iv&£itav TQirriv araAoyo* 

nQOBEVQttV. 



^ 



2. Poat AT add. V: Set ffij T^t> AB «T(*ijtov t^ ^FTttjtij- 
tiivji o(io/(os «(ifi*. tffTM Tfifi>;jt^i'i( fj AF. 4. rB] BF 

Bp, V e corr. m. 2. 6. **'] om. p. 8. HS] MB F, corr. 



ELEMENTORUM LBBER VI. 



107 




Sit data recta linea non secta AB, recta autem 
AF secta in punctis ^, E, et ponantur ita, ut quem- 

Ubet angulum comprehendant, 
et ducatur FJS, et per zi, E 
rectae BF parallelae ducantur 
^Zy EHy et per zi rectae AB 
parallela ducatur ^&K [I, 31]. 
itaque utrumque Z0, 0J5 par- 
allelogrammum est. quare 
JG^^ZH et &K'^HB 
[I, 34]. et quoniam in triangulo ^KF uni lateri KF 
parallela ducta est recta GE, erit FE : E^ «= K& : &^^ 
[prop.n]. BedK& = BH,&J^HZ. it&.q\xe TEiE^ 
= BH:HZ. rursus quoniam in triangulo -^H-E uni 
lateri HE parallela ducta est ZJ, erit E/d : /dA 
= HZ : ZA [prop. II]. et demonstratum est, esse 
etiam FE : EJ = BH : HZ. itaque 

rE:EJ = BH: HZ et EJ:^A = HZ: ZA. 
Ergo data recta linea non secta AB datae rectae 
lineae sectae AF congruenter secta est; quod opor- 
tebat fieri. 

XI. 

Datis duabus rectis tertiam proportionalem inuenire. 



m. 2. 9. xatl postea ins. F. 11. xriv EJ] EJ Bpet in 
ras. F. KGj corr. m. 2 ex GKY. 12. t^v] om. BFp. 

13. ngoQ trjv] ngog BFp, et sic deinde per totam prop. 
15. HE] corr. ex EH m. 2 V. 17. ^] postea ins. F. 18. 
ovTCDff]*m. 2 V. ^aziv aga ms — 20: t^v HZ] postea insert. 
in ras. m. 1 F; mg. m. 2 V. 19. tijv HZ] HZ etiam V. 

20. EJ] corr. ex JE m. rec. P. ngbg JA ovtag bis F. 

rj] ins. m. rec. P. 24. noi^aai] in ras. m. 1 P. 



108 rroixEiaN s'. 

"Eezaxsav at 8o&£t<tai \^8vo s^lQeVai] al BA, AF 
xal xtie&aeav yoavCav nef}ii%oveai xv%ovaav. SeV iS^ 
Tm' B A, A rtQizrjv avaloyov nQoasvQiiv. ix^E^ijO&a- 
eav yaQ inl t« A, E eTjfteta, xal xele&to r^ ^J^J 
& foij ij Bjd^ xal int^evx%a y BF, xaX Sia xov ^^ 
jtapaA^ijAog avtfi ^%ita i} A E. 

'End ovv TQiyfovov tow A^dE naffa (i(av rmv 

nkev(fwv zriv /JE i]«zai ij BF, avdloyov iaTi.v lag 

t} AB Ttpog tijv BA, offTrag ^ AF repog zijv FE, 

10 fffij ^i ^ SA ZTJ AF. ieriv ai/a mg fj AB xgi 

zr}v Ar, QVTiog 17 AT nrpos t^v FE. 

Avo «pa dod-EiCiav av^Emv ztov AB, AT Xfflvt^ 
avdXoyov avtaig jtpoGsvQijTai r] FE' onsff iSst noiijettt. 



aai. 

i 



' Tffimv 8o&siaav av&sttav zszapzTiv 
koyov ngoesvQetv. 

"Eexaeav at So&steai xqstg sv^stat at A, B, 
Sst dij Tiov A, B, r zBzaQzrjv dvdkoyov K901I 

fVQBtV. 

3 'EMxtCe&oeav Svo ev&ttat at AE, /iZ ymvlav 
jiEQtBxovactt [Tvxovaav] TTjv vao EAZ' xal xtia&tii tt} 
[liv A Cerj i] JH, tt^ Sl B tOr} i) HE, xal ht t^ 
r i'er\ 17 J®' xttl im^evx^Bietjg tijg H@ itttgaXXri^os 
KWT^ ^X^^ ^'^ "^o^ E i] EZ. 

6 'EjibI ovv ZQtydvov tov /dEZ naffa [liav trqvg 

1. Bvo fv&ilai] om. P, iv&eiai anpra BCr. m. rec. 3. BA 
e corr. V. tiettv V. 4. yce atJB, ATTheon (BVp? yapW 

Ej,ArF). 6. Br]rBp. s.jeja^- « -i-i^ 

om. BFp. B.d] BA F. AF] A in 1 

njv] om. Bp. TI3.] om, Bp. FE] . 

Biji^e P, corr. m. S. 20. iv.*tle9ia laiv rp (no 




ELEMENTORUM LBBER VI. 



109 



Sint datae rectae BAy AF et ponantur ita^ ut 
quemlibet angulum compreliendant. oportet igitur rec- 
tarum BA^ AF tertiam proportionalem inuenire. 

producantur enim ad punc^ 

ta ^, E, et ponatur AF 

= Bdy et ducatur BF^ et 

, ' per ^ ei parallela ducatur 

• ^E [I, 31]. iam quoniam in 

triangulo A/iE uni lateri 
/lE parallela ducta est BT, 
erit AB\BA = AT \ TE 
[prop. n]. sed 5^ = AT 
itaque AB\AT= ATiTE. 
Ergo datis duabus rectis AB, AT tertia earum 
proportionalis inuenta est TE] quod oportebat fieri. 




XIL 

Datis tribus rectis lineis quartam proportionalem 
inuenire. 

Sint datae rectae A, B, T oportet igitur rectarum 
A, B, T quartam proportionalem inuenire. 

ponantur duae rectae ^E^ ^Z 
ita, ut quemlibet angulum com- 
prehendant E^Z, et ponatur 
z/H = A, HE = B, J® = T 
et ducta recta HS ei parallela 
per E ducatur EZ [I, 31]. 
iam quoniam in triangulo ^EZ uni lateri EZ 




xvxwcav] om. P. xa^] om. p. 

roiv nXBVQmv Theon (BFVp). 



T^] €17 <p. 25. ydav 



110 



ETOIXEIiiN = 



HS, oiiTros ^ ^© jrpos rjjv &Z. fetj dt ^ fiiw ^H 
zfl J, ij 31 HE TJj B, 7i St ^® z^ r- ieriv «pw «tg 
71 A Tiqoq %r\v B, oiirrag ^ F ?rpos f^v &Z. I 

5 Tgimv aga So&€i6dv Bv^stav tmv A, B, P r»fl 

TKpri; KVttAoyov jtpoOfvp^jTat ^ ®Z' o;r£p ^tfft 3toi^ff«i," 



^vo dod^BtSiav ev&tiav fidei}v avaKoyov 
^igoeBvpiiv. 
3 "Eexaeav at 8o%£teat Svo ev&Etai al AB, BF' 
Sii dij Trow AB, BT (teatjv avaXoyov jtpoeevpetv. 

Ksie&mettv iit tv&tias, xal yty^K^pQ^a i:tl t^5 
AV ijfUKVxkiov ro A^F, xal ijj[&a axo To£l B Oij- 
fitiov T^ AF sv&sia stffog OQ^iis ly B^, xal iat- 
16 ^svx^meav aC A^, ^F. 

'Ejtel iv i}(ttxvxii^ yavCa ierlv ^ vao A^Pj 

og&i^ iSTiv. xal ixil iv OQQoyiavia} ZQiyav^ tm 

AAF ajto T^s dpd^ff yavias i^^ ^V'" ^^eiv xd&szog 

rpizat ij AB, ij z/S a^a tmv f^s ^isias Tiitjndtav 

> tav ABy BT fteojj dvttXoyov ieziv. 

Avo Kpa do&tiBiHv £v&£tav T<av AB, BF lUeTj 
avttXoyov nQoeavQtjtai i) AB' ojTEp ISst xot^eai. 



i3'. 
l leoyaviav naQalXfjXoi 

la g n. XIV. Phil J 



Tmv teav ze ; 



1. £ZJ corr. ox H® m. rec. P; H@ Bp. H©] . 
ei ZE m. rec. P; EZ Bp; 0H V m. 2. ij] om. V. 
in raB. B. i^»} ow. BFp. 2. tt)'»] om. BFp. 

e corr. Vj ze P. 4. ©Z] Z in raa. F; ZS P. 



ELEMENTORUM LIBER VI. 111 

parallek ducta est H®, erit ^H: HE «= ^® : ®Z. 
sed ^H «= Ay HE «5, ^® = T. itaque ^ : B 
^r.QZ. 

Ergo datis tribus rectis lineis A, B, F quarta pro- 
portionalis inuenta est ®Z; quod oporteW fieri. 

xm. 

Datis duabus rectis lineis mediam proportionalem 
inuenire. 

Sint duae rectae datae AB, BR oportet igitur 
rectarum AB, BF mediam proportionalem inuenire. 

J ponantur in eadem recta, et 

in ^r* describatur semicirculus 
A^jr, et B, B puncto ducatur 
ad rectam AF perpendicularis 
A 5 T B^, et dueantur A^, ^F. 

iam quoniam in semieirculo est L AAF, rectus 
est [III; 31]. et quoniam in triangulo rectangulo 
A^r a recto angulo ad basim perpendicularis ducta 
est ABj AB partium basis AB, BF media propor- 
tionalis est [prop. VIII coroU.]. 

Ergo datis duabus rectis lineis AB, BF media pro- 
portionalis inuenta est z/J3; quod oportebat fieri. 

XIV. 
In parallelogrammis aequalibus et aequiangulis 

^bIo] om. Bp. 16. %al insi V. 19. JB'] BJ F; V, corr. 
m. i. JB] BJ Y, corr. m. 2. 21. fisariv P, Bed corr. 

22. nQOiniv^tai, F. 24. ts] om. p. xat] m. 2 F. lao- 
ymv^av] P, Philoponns; ikiav (ua i^ariv ix^vxmv ynvCav Theon 
(BVp; in F om. y,Cav et snpra scr. /yica seq. ras. 1 litt.), P 
supra m. rec. 





112 STOIXEIP-N ?■. 

yQa[i(i(ov dvzt.ncx6v&aetv aC alsvpal ai aEQl 
taq taaq y(avia<^' ■aaX cov looytavimv jrapaAAij- 
l,oyQKfi.ficov avriasaov&aetv at ■xkEvgal at 
nsQl ruq fs«s yavCaq, iOa iazlv ixftva. 

6 "EOza taa T6 xal leoytovia jtaQall7ji.6yQa(i[ta ta 
AB, BF iaag tj^ovta rag rcpoff ta B ycaviag, xal 
XEie%iaeav iit' Bv^sias at ^B, BE' iit Bvd-eiag 
apa dal xal al 7.B, BH. Uya, oti tdv AB, BF 
RvtiTtsnov&aatv aC JtXfvpal aC %€qI tag [Gag yaviag, 

10 Tovtiettv, oTt ietlv rag ^ /iB irpog t^v BE, ovrmg 
5j HB npbg t^v BZ. 

2^v[ixfa^i]Qt6e&ci y«Q zo ZE KaQakl.-qX6yQay.fiQv, 
ixEl ovv i'0ov ietl tb AB naQaXXrjXoyQafifiov tm 
Br naQaXXjiXoyQafifim , «kXo 6e ti zb ZE, ietiv 

15 «pa tos to AB nQog zb ZE, ovtojs to BT npos tb 
ZE. ttXX' ag (liv tb AB TTQbg tb ZE, ovta^g ij jdB 
itQog TTjv BE, ag d^ tb BF ntQog to ZB, oufrag ij 
HB jrpog i)j'v BZ' xal ag «p(( i} ^B «pos Tqv BE, 
ovtag Ti HB irpos trjv BZ. zav «pK AB, BF reap- 

20 aXXTjXoyQaiifiiov avtntenov&aaiv aC jtXsvQal aC xsqI 
rag Caag yavCag. 

'AXKa Sij leta mg ij .dB ;rpog zijv BE, ovxtag ij 
HB jrpos Tijv BZ" Xiym, oti Haov iatl ro AB aaff-_ 
aXK^qXoyQayyov rt5 BF naQaXXj^XoyQaitiia. aM 

25 'Exd yaQ iaziv vtg q ^B JiQog r^i' BE, ovToifl 
il HB HQOS tiv BZ, alX' dtg lilv ij JB JtQbg vfyf' 



2. ttotiavCiav] om. Theon (BFVp); del, m. ree. P. Post 
irapoHlTjltoypo^ifiiDV add. Theon: ^Cav yiavlav (tii ymiiij! lariv 
Ijov-ciov (BFp; /llav fiia fffijv ixovimv ycavCav V). 6. zt 

kbI iBoytivta.] om. Theon (BF?p); del. m. reo. P. 7. *el- 

(f*» V. 8. tUCv PBp. 10, iffTfV] om. p. rjji'] om. 



ELEMENTORUM LIBER VI. 



113 



latera aequales angulos compreliendentia in contraria 
proportione sunt; et parallelogramma aequiangula; 
quorum latera aequales angulos comprehendentia in 
contraria proportione sint, aequalia sunt. 

Sint aequalia et aequiangula 
parallelogramma AByBF aequa- 
les habentia angulos ad J3 po- 
sitoSy et ponantur in eadem 
recta dBy BE. itaque etiam 
ZB, BH in eadem recta suni 
dicO; in j4B, BF latera aequa- 
les angulos comprehendentia in 
contraria proportione esse^ h. e. 
esse jdB:BE = HB : BZ. 

expleatur enim ZE parallelogrammum. iam quoni- 
am AB = BFy et alia quaedam magnitudo est ZEy 
erit AB: ZE ^ BF: ZE [V, 7]. sed AB : ZE 
= ^B:BE [prop. I], et Br:ZE~HB: BZ [id.]. 
quare etiam ^B : BE = HB : BZ, itaque in parallelo- 
grammis AB^BF latera aequales angulos comprehen- 
dentia in contraria proportione sunt. 

iam uero sit AB : BE = HB : BZ. dico, esse 
AB = Br. 

nam quoniam est JB : BE = HB :BZ,et JB : BE 




BFp. JBE] corr. ex B9 m. rec. P. 11. rijv] om. BFp. 

BZ] ZB P. 12. ZE] EZ p. 17. t»}v] om. BF; ro p. 
TO ZE] ZE BF; Z in ras. m. 2 V. 18. WQOff tjv] nqoq 

BFp, et sic deinde per totam prop. mq aqa] cnansQ V. 

^ JB] BJ p. 19. a^a] supra m. 1, sed post BrP. 22. 
dXXa Srj] in ras. m. 1 p. Post di^ add. Tneon: uvTinsnoV' 
^ixmaav at nXsvqal at nsql tag i'aag ymvCag Y,d£ (BFVp). 

28. BZ] ZB V, laxCv P. 26. t?}i/] corr. ex t^ m. 2 V. 



26. coff] e corr. F. ^] om. F. 

Euolides, odd. Helberg et Menge. II. 



8 



114 STOIXEian S'. 

BE, ovTcag to ^B naqaXkfiXoyQamiov sipog ro ZE 
7taQttki.tiX6yQttfi(iiiv, atg di i] HB srpog tTjv BZ, ov- 
TQjg ro Br jtttpttklTjkoyQaiifiov ZQog ro ZE JlUQaX- 
Xrji.6ypafL{iov, xal ais apK zo AB sipog rh ZE, ov- 
6 rag to BF jrpos to Z£' l'Sov aga iozl rh AB na^- 
«XXtiXoyQaftfiov za BF xaQaXXrjXoyQafifi^. 

Tmv apa taistv te xal teoyavCav TcaQaXXTjXoyQafL- 

fiav Rvzmtnov&aSiv al itXsvpttl af jrepi zag ttJag 

ytavCttg' xal a>v ieoyavCav }ta(/aXXijXoyQafi(iav avt^ 

10 ittitov&aSiv aC itXtvpal at mql xag Haas ycavCas. 

tezlv inBlva- omp edei det^ai. 



Tfov i!0cov xal (littv fi.i.tt feijv ^jo' 
vittv XQtyoavaiv avziiiBaov&aatv at hXbvq: 
15 at xsqI tas Csag yatviag' xal mv fiiav fii^ 
i!0jjv ij^ovzeov ycaviav zpiyavatv uvTiiCdXov- 
&at}tv at nXevQal at tzbqI rag fffag yaviag, 
i'Ga iezlv ixetva. 

"Eeza i'0a tqiyava ro: ABF, A^S fiiav fua larfv 

20 £j;oi/ia yavCttv ztjv vjto BAF Ty vno ^AE' Xdyat, 

Stt tav ABF, AAE xQtymvmv dvtiJiBitov&aaLV aC 

xXsvQal at xbqI zag taag yavCag, Tovriaziv, Sti iazlv 

ag 71 FA JtQog ftfv AA, otSrros rf EA irpoe t^i 

KBCeQa yag aGte in ev&sCag slvai tip/ VA 

25 AjJ' i%' ev^^eCag uQa icrl xal if BA t^ AB, 

i7tti,Bvx&Ci 71 BA. 



] 



1. npss To — 2: cos Si~\ iofiert. in las, F. 2. jcaqaj.ina 
Xoy^a^^ov] om. V. 3. ZE KasaXlriloy^a^iiov} V\ ZE TheM 
(BFVp), 5, inzCv P, eomp. p. 7. iaa,v a^a p, h] od| 
Bp. taoyiaviBiv] PBPpj in P Hnpra scr. m, rec. roij» yet 
vCav fiiav ^ta fjoirimvj fi^av ftta Carjv ixovitov •/mvlav V, Sfl 



^V _ ELEMESTOEITM LIBER VL 115 

= AB : ZE, UB : BZ = Br: Zfi [prop. I], erit etiani 
^B:2E=Br:Z£;[V,ll]. itaque ^B = Br [V,9]. 
Ergo in parallelogrammia aequalibas et aequi- 
aDgulis latera aequalea anguloa compreheudeiitiia in 
coutraria proportione sunt; et parallelogramma aequi- 
, angula, quorum latera aequales augulos comprehen- 

Ftia in coutraria proportione sint, aequalia suut; 
id erat demonstrandum. 
XV. 
In triaugulis aequalibus, et qui unum angulum 
um aequalem habeant, latera aequales angulos com- 
prehendentia in contraria proportione aunt; et trian- 
guli unum angulum uni aequalem habentes, et iu 
quihua latera aequales angulos comprehendentia in 
contraria proportione sint, aequalee sunt. 

, Sint aequales trianguli ABF, AJE unum augu- 
lum imi aequalem habentea, LBAF = /JAE. dico, 
in triangulis ABF, AAE latera aequalea angulos 
comprehendentia in contraria proportione esse, h. e. 
6886 rA:AA = EA: AB. 

ponantur euim ita, ut VA et A^ in eadem recta 
• eint. itaque etiam EA et AB in eadem recta sunt. 
■et dncatur BA. iam quoniam AABr= A^E, et 



hiati fua punctiB deL 9- laoymvitor jraeoWiiloycKfifiw»'] 
PB, F (poa' ' " 



, _ (poat iao- ras. 1 litt.), p; in P m, rec. aupra scr, roij* 

ytotiav (i^av fitfi Ixoviaiv; liiav fiiS (punctJB del,) i^ativ Ixov- 
lOjv ytaviav jiaetfilijitoyeOftfKn»' V, 16. «f] m. 2 P. mv 

ipiywvw* F. 16. tQiyioviov'} oni. FV. 20. rpl corr. 



rec. P. Ifyto, oti] et seq. inBert, 

' Jttgi] Bfe^ P, corr. m. 2, 23. wpoff i^'vl bifi Jepos BF' 

21. rA} jry,Yia ra», 26. htiv PBF, oomp, p. 



( 



22. M 

J 



i 



116 ETOlXEIiiN 5". 

'Ejtst ovv laov itstl t6 ABT tQCymvov rp AME 
tQiymva, alXo 6i ti to BAd, Igtiv apa (05 ro 
rAB tQtyiavov itffbg tb BA/i tQiytovov, ovzias to 
EA/d TpCytavov aQog ro BAd tQCyavov. uXk' mg 
E ^\v xb FAB zgbs ro BA^, omms ij PA jtpos t^v 
A^, mg tfi to EAA Bpog to BA^, ovrag ^ EA 
itgos ti}v AB. xal 005 aga ij FA spog t^v AjH, 
ovtatq 7] EA nrpog r^v AB. tav ABT, A^E «pK 
XQiyeivmv dvTizeitov&aaiv al skEVQal «t ateqX 

10 foag yavCitq. 

'Akla 6i} avttmsnov&hmiSav aC JtXBVQal zav AB. 
AAE TQiydvtov, xal Idtw dtg 17 VA itpbs trjv A. 
ouTMg ii EA Jfpos Tijv AB- kiya, ort fffov 
ro ABF tQiyiovov ta AdE TQiydva. 

15 'Em^Ev%&£(aTiq yccQ aaktv tijs BA, imC ieti 

71 VA JtQos trjv AA, owrws 17 EA arpog t7\v AB, 
akk' ag liiv ij FA Jtpog rijv AA, ouros ro ABF 
tQCyiavov jcgbs ro BA.£t zQCyavov, ix>s dl ij EA nQos 
Ttjv AB, ourrag ro EAA tgCymvov ;tpos ro BAA 

20 TQCyctvov, tog aga ro ABF TQCyatvov Jipos to BAA 
zQCyasvov, ovtag to EAA ZQiyavov Jtpog ro BAA 
XQCyiavov. ixaTEQOv UQa Tav ABF, EAA npog ro 
BA.A tov Kvtox' ^x" koyov, taov apa ^ffrl ro AB, 
ItQCyatvov'} Ta EAA ZQiytovp. 

25 Tav aQtt Hetov xul (iCav (iia lativ i%6vTtov yavi 
tQiyavtov dvrtaeaov&aeiv al akivQtil at «eqX 
taag yiavCas' xal av (iCav [iia tanv i%6vtiav yavCav 

a. 11] om. BFVp, BAd] in ras. m. 2 T, -. _- 
"T"A-BF: BAr Bp,Y m. 2. ovras] ovzm P, oStms aea ^- . 
4. EAJ] BFp, Vm. 2; ^/)E V m. I; JAE P. B^J] litt. 
BA m ras. in. 2 V. tQiyuiio*'] comp. V. 7. t^f] (piiiu] 



! acct F. 




^< 



ELEMENTORXJM LIBER VI. 117 

2< alia quaedaiu maguitudo eet 
BA^, erit ATAB -.BA^ 
= EA^ iBAJ [V, 7]. sed 

: JA et £^^ : BAA 

= £^ : -^B. quare etiam 

FA-.AA^^ EA -.AB. ita- 

que triangulorum ABr, 

/lE latera aequales auguloa comprehendentia iu 

coatraiia proportioue sunt, 

iam uero latera triangulorum ABr,AAE in coQ- 
iraria proportione sint, et sit rA : Ad = EA : AB. 
3, esse AABr= AAAE. 

ducta enim rursus BA, quoDiam eat PA : A.d 
= EA : AB, et FA : AJ = A ABT: A BA^, et 
£--^ :AB = AEAA:ABA^ [prop. I], erit AABF 
: A BAJ = A £.4^ : ABA^. itaque uterque trian- 
gulus ABFjEAJ ad BAA eandem rationem habet. 
quare A ABr= A EAJ [T, 9]. 

Ergo in trjangulis aequalibus, et qui unum angu- 
lum uni aequalem habeant, latera aequales angulos 
compreheudentia in contraria proportione aunt; et 
trianguli unum angulum uni aequalem babentes, et 
in qnibuB lateia aequales asgulos comprehendentia 

corr. ei tov m. 1 F. 8. a^u xijiyaiviav'^ xfjtyiavav a^a V; 

&^ yaiviav p. 13. ipeymviov] fBiviiv p. Ag] poatea 

iuHert, m. 1 P; om. F. mpoe thv] xfoe BFp, et sic deiude 
per totam prop. 16. FA] AF p. 19. tij*J om. etiam V. 
20. ABT} BAT P. Poet xQiymvov add. F: ovtms lo EA.d 
Ttfymvov, sed del. m. 1. 21, tpiyoivov] om. V. ovtiu;] 

om. F. 10 EAJ tfiyaivov «Qoq to BAJ tffiyatvov] om, 

BFp. 22. cpo] om, Bp. 23. tativ P, comp. p, 24. ip^- 
ymvov] om. ?. 26, itXevQal at] om. F. 27. yiavias Jtlevi/tiiF. 






TQiymvmv avrmexov&afftv ttt nkEVQoX 

taag yaviag, ixiiva Ciftt ifStCv oitEQ ISai Sst^i 

'Eav tdeeapts evQ^stai avaloyov eoaii 
b v%o rmv axQiov iteQiexofievov otf&oyioviov fSoit 
iarl xm VTto riBv (liaav niQiexoiisvm op#o- 
yavip' xav to V7l6 tmv KXt/tov neQtex6fi,EV0li 
OQ&oytoviov Haov ^ ta vnb tav fiitlmv iteQii 
Xoiiiva OQ&oycovia, at tiaauQts ev&etai avt 
li> koyov eaovzai. 

"Eazaeav tianaQes ev&eUu avdXoyov ai AB, F^^ 

E, Z, cas ij ^B ^Qog ttiv r^^, ovTag j^ E icphg 

TTiv Z" Xiya, ozi z6 vitb tav jiB, Z nEQiE%6[i,evov 

oQ&oyaviov i'aov iatl t^ vno tav T^, E TtBQisxof 

15 (iiva oQ&oyavCa. 

"Hx&taeav [yaQ] ano Trov ^, V arjfiEiaiv raCg AB 
FjJ ev&eiais ra^og opOig aC AH, F®, xal xeCa&ti 
ty fiev Z i'ati i\ AH, tri 61 E i'arj r} F®, xaX av(i> 
aeicXriQaa&a za BH, A& TtaQaXXrjXoyQafifia. 
Kal ineC iativ as ij AB XQog tijv F^, oikois i 
E TtQos tiiv Z, tai} 6i ■{} fiiv E ttj r&, ^ ffi Z i 
AH, ^etiv apa ag ^ AB repog tijv VA, oilzais ( 
r@ Jtpog r^v AH. zav BH, jJ@ UQa itaQttXXtjlo^ 
yQaftfiav avzixeaovQaeiv at nXevQal «t JtEQl ra^ 
35 taag yavias. ov 6t taoytavCav saQalXijXoyfianfitoii 
avttmenov&aaiv at nXev^al at itEQl tag laaq yaivlat^ 



2. iaziv] ilaiv V. 4. a,ni PBp. 7. KiCvl Hffl tl % 

11. nE ziava^es P. dvcHayov~\ om, V. 12. Z avaXotov "9 
Tij»'] oiQ. Bp. 13. AB] B in cas. m. 2 V. Z] eraa "^ 

11. caiiV P, comp. p. E} poatea add. m. 1 p; era8 



ELEMEOTORUM LIBER VI. 



119 



in contraxia proportione sint, aequales sunt; quod 
erat demonstrandum. 

XVI. 

Si quattuor rectae proportionales sunt^ rectangulum 
extremis terminis comprehensum aequale est rectan- 
gulo mediis comprehenso; et si rectangulum extremis 
terminis comprehensum aequale est rectangulo mediis 
comprehensO; quattuor rectae proportionales sunt. 

Sint quattuor rectae proportionales j4B, Fz/, E, Z, 
ita ut sit j4B : F^ = E : Z. dico, esse AB X Z 

^r^xE. 



H 



S 



B 



Eh 



r 

Zi 



ducantur a punctis -^, F ad rectas j4B, Fd per- 
pendiculares AH^ F®, et ponatur AH = Z et F® = E. 
et expleantur parallelogramma BH, ^t&, 

et quoniam est AB : Fd = iS? : Z, et ^ = F®, 
Z = AH, erit AB : FA = TS : AH. itaque in 
parallelogrammis BH, A® latera aequales angulos 
comprehendentia in contraria proportione sunt. paral- 
lelogramma autem aequiangula, quorum latera aequales 
angulos comprehendentia in contraria proportione 



16. yap] om. P. 18. av(insnXriQ<6a^<oaav BFVp. 22. AH] 
corr. ex AJm, rec. P. 28. AH'] post ras. 1 litt., H e 

corr. V; corr. ex A@ m. rec. P. apa] m. 2 V. 24. at 
'nsQQ nsQ^ P. 



120 ETOOEIiiN S'. ■ 

£aa iatlv exelva- Hoov a^a ietl ro BH %aQak}.r\X6- 
yganiiov ta z/0 itapaAlrji.oy^anfi^. xaC iezi ro 
^\v BH to vno tmv AB,Z' f6)j ya^ ^ AH i^ Z- 
zb &t jd® to vno rwv r^, E' tarj yag ij £ tjj r&' 
5 To aga vxh tav AB, Z jteQitxoficvov oQ&oywviov 
tsov istl ta vao tav T^d, E mQiti%o^ivij> 6(/&oy(ov£a>. 
'Ai,la Si} ro vjrtJ tmv AB, Z TttQttxoy^^vov dpfro- 
ytaVLOV taov lazm tm vito zmv ^A, E JtBQiexofi^v^ 
d(}9oyavia' Xiya, oti at tiasapts tv&Etai dvdloyo^ 

10 SffovTai, mg 1/ AB ngbs zijv FA, ovzmg 17 E Xfi 
tijv Z. 

Tmv yag avzmv xataexEvae&ivziov , iittl to vi 
ziav AB,Z leov iati ta vitb rroi' FA, E, xaC it 
to fiiv vito rtoi' AB, Z lo BH' fffTj ydg ietiv 

IB AH ty Z- ro dl vitb tmv VA, E zb A&' fffij yat 
il r& r^ E- tb KQa BH i'eov ietl rra A@. xai leziv 
teoymvta. zav Si teav xal ieoymvCmv napall^^Jio- 
ypd[iiimv dvtiTtEaov&aaiv aC nktvQal aC Xtpl tdg teag 
ymvCag. teziv a^a (Off 17 AB Ttgbs rjjv F^, ovtcag ij 

30 r& jrpos rijv AH. tei) di ^ {iiv T® ty E, 17 Sh 
AH zf) 2" lativ Sffa (Dg rj AB repog zr^v r^, ovti 
r} E Jtpog zijv Z. 

'Edv dga tiaeaQts iv&Etat dvdAoyov meiv 
vjtb zmv dxgmv TttQitxo^itvov ogd^oyiDviov teov ietl 

25 tp vxb tinv (liamv itEQiExoiitvp oQd^oyiovip' xav zo 
vjib zav dxQmv atQiexojitvov 6Q&oy<avi.ov taov 
VTtb tmv [lieav atgit%oiJiiv^ op^oymvCp, al tieea[ 
ev&ttai dvdXoyov ^aovtai' oxeq ISet Stt^a: 



i. r^, E] seq. scefifxoiievov oa^oyiaviov V, pnilctiB del 
E] corr. ei r& m. 2 V, rS] cort. ei E m. 2 " 



iv^ 

I 



1 



ELEMENTORUM LIBER VI. 121 

sint, aequalia sunt [prop. XIY]. itaque BH= jd@, 
et BH= ABX Z (nekm AH=Z) et ^& = rJxE 
(nam B = T®). itaque ABxZ = FA X E. 

iam uero sit ABxZ = Fjd X E, dico, quattuor 
rectas proportionales esse, AB : Fjd = E: Z, 

ham iisdem comparatis, quoniam AB xZ = T^ 
XE,etABxZ = BH (namAH^Z), et F/IX E 
= AB (nam T® = jB), erit BH — jd®. eadem 
autem aequiangula sunt. et in parallelogrammis 
aequalibus et aequiangulis latera aequales angulos com- 
preliendentia in contraria proportione sunt [prop. XIV]. 
itaque AB:rjd = r&: AH. sed T® — E, AH= Z, 
quare AB : F^ = E:Z. 

Ergo si quattuor rectae proportionales sunt, rectan- 
gulum extremis terminis comprehensum aequale est 
rectangulo mediis comprehenso; et si rectangulum 
extremis terminis comprehensum aequale est rectan- 
gulo mediis comprehenso quattuor rectae proportionales 
sunt; quod erat demonstrandum. 



9UQi6xoikiv€9P oq^oyoiviaiv F, sed corr. 8. Tooy] mutat. in 

tmi F. 9. oq^oyoavCoiv F, sed corr. 14. icxiv\ om. V. rj 
AH t^ Z] Tjj Z ^ ^H V; in F m. 2 ex trji Z fecit t^ HZ, 
16. tcri yaQ 17 — 16: tm z/(9] mg. m. rec. P. 16. iativ^ P; 
slaiv BFVp. 19. ^] (alt.) postea ins. m. 1 p. 20. r (9] 
corr. ex HS m. 1 p. AH] corr. ez ZJf m. 1 p. 23. 

aai, PBVp. 26. x«v] xal ci V. 26. ^] iati F. 27. 
tiaaaQss] seq. ras. 2 litt. F. 




rpffff 



122 ETOIXEIiiN S'. 

'£ci/ TpeTs evQ^Btai avaAoyov ojiji 
TKiv axgav xeqiexoiievov OQ&oym 
ierl xm a^o r^g ^aOtjg Tsipayava' 
5 ^310 xav ttxgav 7tdQiex6(iBvov oQ&oyavtov toav 
fj T^ dnb T^s fiBgrjg rsTQttyav^f aC Tpffff 
evQ^etai dvdloyov iffovTai. 

"EdTaOttv TQstg ev&stca dvdloyov al A, B, f, 

il A Ttpog TijV B, ovrcag 17 B apog t^v f' 

10 oTi t6 wjto Tiov A, r neQiexoC^BVov 6(^oydvt.ov 

iffrl ta dxo Tijg B TBTpteyoiva. 

Ks(6&a Ti5 B /ffij ^ J. 

Kal izsi ioTiv ag ri A apog t^f B, otSrias 55 

B Jtpo? nji' r, fffi; S\ ^ B tf) d, Sartv &Qa ag ^ 

15 A JTpog T^v B, ri A «qog t^i/ Z'. ^av Sl rieaaQsg 

evQ^sTai dvd^oyov deiv, to vxo tmv SxQav %sqie%6- 

{isvov [oQQoydviov] teov iffr! Tt5 wcb Tav fiitsav 

nsQiexoiiiv^ OQ&oyavia. t6 apa iwo Ttov ^, F i'ffof 

^ffil rro wrco tdv B, -^. «A^ii To vno rdv B, jd t6 

20 d%6 tJig B sStiv leri yaQ »j .8 t^ A' t6 UQa imo 

rmv A, r re£ pif xofifvor OQ&oydvtov taov darl xp 

KJTO Tijg B TSTQaydva. 

'Alla Sri r6 iito twv A, F taov iata tp i«o 
tije B' liya, ort ^ffTiv dg -^ A JtQog tjjv B. 
2b T] B ^Qog Ttjv r. 

Tdv yuQ avtdv xataaxevaa&ivrav , ineX 
tdv A,r fffov iaTl rd dao t^s B, ai,i.a 
r^S -8 To vito T(DV B, A iattV tat] yap 57 B r^ 
t6 aQa V7t6 tdv A,r taov iatl td t«r6 tdv B,^. 

1. ij'] et litt. iuitialie m. 2 V. 2. ds( coda. 4. 

KQv] xcti tl Y. 8. T^s] inBbrt. poBtea F. S. aC Tftic P. 



ELEMENTORUM LIBER VI. 123 

XVII. 

Si tres rectae proportionales sunt, rectangulum 
extremis terminis comprehensum aequale est quadrato 
medii; et si rectangulum extremis terminis compre- 
hensum aequale est quadrato medii^ tres rectae pro- 
portionales erunt. 

Sint tres rectae proportionales A^ B, F, ita ut sit 
j4 : B = B:r. dico, esse Axr= B\ 



A^ 



B\ 1 J\- 



ponator jd = B. et quoniam est A : B = B: Fy 
et B = jd, erit ji : B = ^ : F. sin quattuor rectae 
proportionales sunt, rectangulum extremis terminis 
comprehensum aequale est rectangulo mediis com- 
prehenso [prop. XVI]. itaque A X F = B X jd. 
uerum Bx ^ = B^; nam B = J, quare 

Axr=B\ 

iam uero sit Axr= B^. dico, esse A:B =B:r, 
nam iisdem comparatis, quoniam AxF = S^, 

ei B^ = BXA (nam B = A\ erit A X r= BxA. 

sin rectangulum extremis terminis comprehensum 



XVII. Philoponus in Arist. de anima g II. 

12. ns^cd^oa ydg P. J] post ras. 1 litt. F. 16. iat codd. 
17. oQd^oycoviov'^ om. P. 19. B, Jj (prius) in ras. m. 2 V. 
aXXd — B, J] insert. m. 1 F. 20. iariv Pari] eras. F. 24. 
A] B n. 26. insQ corr. ex ini m. 2 V. 27. dXla to dno 
xfig B xo vno tmv B, J iaxiv] PBp; idem, sed xm vno V, 
F mg.; xovtiaxiv tm vno tav B^ J F, 28. tari] -ij m ras. B. 
T^ Jj in mg. transit m. 1 V (supra est ras.). 



124 ETOIXEias s'. 

ittv dl ro vno rmv axgav fffoi' jj ta imo xav (if- 
acov, aC tiseaqes Bv&stat avaloyov eletv, lottv Spa 
(hq ii A wpog T^v B, ovzcog 17 ^ «pos t^v T. f<nj 
di 7) B zf) ^' as Kp« 17 .^ JTpog r^v B, oirrcDS ^ J 
5 Ttpog tijv r. 

'Eav UQa tfitts sv&siat avdXoyov aUiv, 
tdv ttXQav xtQisxotisvov og&oyavtov Isov ietl rp 
{tatb tijs itiotis tetpaytovp' xav ro vno tiov axQwv 
jctpisxoiifvoi' og&oydvtov tSov jj rra awo t^g (iiffi 
10 zstQuyoiva, aC TpEfg tv&ttai avukoyov ieovtaf 
iSti Sfi^tti. 

U]'. 

'^no tijs So&^eierjs sv&Eias za do&ivn 
£v&vyQdft(ta oftotov te xal oiioitas xctfievo! 

15 £V^vyQafiiiov avaypailrai. 

"Eezsi i} fiiv So&ttOa ev^BZa ij AB, zb 8h io#iv 
ev&vyQttfifiov tb FE' Sit dij ajtb t^s AB ev&ei 
ta VE sv&vyQd(t[ia ofioiov zs xaX ofioCiog xeifiBt 
ev&vyQttfiiiov dvaypd^^at, 

20 '£jre£swz^o) ij -JZ, xal evveatdtia n^bg ttj 
fd&siu xal ToEg npos avtfj et][uiois toEs ^,B rij 
fihv wpos Tp r yatvi^ tatj ij uwo HAB, zy Sh vnb 
T^T. tatj ij vao ABH. A01W17 «p« ^ r^ro /'Z^ t^ 
ujro AHB iettv terf' iaoyiaviov «pK ietl tb ZPjA 

26 TptVojvov ira HAB tQtycavat. dvdXoyov d^a iozlv 
as ij Zii 3ipog f^v HB, ovTrog ^ ZF jrpog z^v HA, 
xk\ ij FA Jtgbs iijv AB. ndliv ewsatdta nQog 
rfj BH tvO^tia xal torg itpos avtfj erifisiots rofs B, 



1 

ov (16- 
&lf« 

te^ 
■\ tp 

Xp(DV 

"'fflK— 

EV0I9 
o»iv 

7 -^bB 



i PFVp, 7. ^oi/v P. 8. TiSv — 10! fooi 

9. 3] iotC cotop. F, Eupra acr. 5. 



^^i^H 



ELEMENTORUM LIBER VI. 125 

aequale est rectangulo mediis comprehensO; quattuor 
rectae proportionales sunt [prop. XYI]. itaque 
A:B = ^iT. sed B = ^. itaque A:B = B:r. 
Ergo si tres rectae proportionales sunt, rectan* 
gulum extremis terminis comprehensum aequale est 
quadrato medii; et si rectangulum eztremis terminis 
comprehensum aequale est quadrato medii^ tres rectae 
proportionales erunt; quod erat demonstrandum. 

XVIII. 

In data recta datae figurae rectilineae similem et 
similiter positam figuram rectilineam construere. 

Sit data recta AB et data figura rectilinea FE. 
oportet igitur in recta AB figurae rectilineae FE 
similem et similiter positam figuram rectilineam 
construere. 

Zr 





A 

ducatur ^Z et ad rectam AB et puncta eius A, B 
angulo ad F posito aequalis construatur L HAB^ 
angulo autem FAZ aequalis L ABH [I, 23]. itaque 
L rZJ = AHB [I, 32]. quare A ZFJ triangulo 
HAB aequiangulus est. itaque ZA : HB = ZF: HA 
= FA : AB [prop. IV]. rursus ad rectam BH ei 

^fioiag n (non P). 20. z/Z] Zz/ P. cvvsaroTO n (non P). 
22. Tw] rfi P. taril om. V. HAB^ BAH P; AB F^ 

HAB tarj V. 23. foi?] om. V. tfj] Xomi t^ V. 24. 

AHB] A"B'H F. hzi] om. V. 26. cos] supra F. 28. 
T§] corr. ex zrig m. 1 p. BH] H snpra acr. V. 



126 ETOlXEIiiN ff'. 

H xfj jilv vm ^ZE ytavCtf larj ij vxb BH&f tij 
6i imh Z/iE fiJij ^ v«o HB0. Xoixij Kpa ij jc^og 
ra E koijtfi rfi tiqos ro & ionv i'6^' feoytoviov a^a 
i<jTl 10 Zz/£ TQfymvov rp H@B TQiyava' avaXoyov 
6 KQa fOzlv (og rj ZA Jrpog r^i/ HB, oiirojs ^ ZE 3cpog 
T^v H@ xai r^ E^ nQog triv &B. iSeix^Tj 6% xtd 
dtg ^ Z^ Tpo? tijv ifB, oOtos ^ Zf npoj r^v i/^.<< 
xal T] r.d «pos T^v AB- xal ms apa rj ZF itQog 
T^v AH, ouTtog ^ « Pz/ Hpos tijv AB xal ij ZE 

10 jtpoj r^v if© xal Iri t) E^ Jipoff njv ©B. x«l 
^jtsi fffjj ^orlv ij (ilv vao TZd ymvia t^ ujri AHB^ 
Tj 6i vab idZE tfi vnb BH&, SXf) aga ^ VTib FZE 
oAij zfi vao AH& itSTiv /ffij. dia za avTa Si\ xa\ 
^ ijjro r^E zfi intb AB& ^ffTtv ffij;. lOTt iSi xal i} 

15 ;*iu rcpog Ta F r^ Jtpog rp A 107] , ii S% jtpog rto £ 
t^ jipog r^ 0. ieoytoviov aga iezl to A& tp PE' 
xal tag )t£pi Tag fffag ytavlaq avtmv nlevQa^ avaXoyov 
dxtf oftotov apa ietl i6 ^® EUiSTJjipaftfiov i 
BV&vyqa^yi:Gi. 

20 !/f]r6 i^s So&eieris apa ev&eias T^g ^^ tcS m 
■&£j'T( Ew^vypaftfioj Toi I^E oiioiov te xal ofioiiog t 
^evov ev&vyQKfi^ov avaydygaitTai ro A&' ojtGp iSM 
zot^0ai, 

25 Ta oftoicc rpt^iova Jtpog «lki]lK iv Siai,^ 

eiovi Xoya ietl t£v ofioXoyav %lev(/mv. 



XITi coroU. FhitoponuB in ana]. poat. 117*. Faelliu p. I 

1. BH&} "B'H"'& F. 2. vxo] om. Bp. fffij] oni, B. 

4. H9B] PF; HB0 B, 7 e corr. m. 2, p corr. ex H9S 

m. 1- 5. Z//] ^Z P. ZE] in tas. ro. 2 V. 6. H»] 



W ELEMENTORDM LIBER VI, 127 

puncta eius S, H angulo dZE aequalis construatur 
L EH& et angulo ZJE aequalis L HB& [I, 23]. 
itaque qui relinquitur augulus ad E positus, reliquo 
angulo ad & posito aequalis est [T, 32]. itaque AZzJE 
triangulo H&B aequiangulus est, quare Z.^ : HB 
= ZE : H& = EJ : &B [prop. IVj. demonstrauimua 
autem, esse etiam Zz/ : HB = ZF: HA = F^ : AB. 
quare etiam Zr:AH=rA: AB = ZE:H&^E^ 
: &B. et qnoma,mLrz A^AHB,etL^ZE = BH&, 
erit L PZE = AH&. eadem de causa etiam L rJE 
= AB&. et praeterea augulus ad F poBitua angulo 
ad A posito aequalis est, et angulus ad E positus 
angulo ad & posito aequalis. itaque A& aequiangula 
est figurae FE. et latera, quae aequalea angulos com- 
prehendunt, proportionalia habent; itaque figura rec- 
tilinea A& similis est figurae rectilineae FE. 

Ergo in data recta AB datae figurae rectilineae 
rE similis et similiter posita figura rectilinea con- 
Btrncta est A®^ quod oportebat fieri. 



SIX. 

Similes trianguli iater se duplicatam rationem 
labent quam latera correspondentia. 



fc: 

fd in raa. m. 2 V. ©B] B9 P. nai ^ EJ neos Tijv <9B] 
■biB F, eed corr. 7. ? le ZF P, 8. xal ros opn — 9: «^* 
AB} om. p. 10. EJ^ "J'E F. 12. JZE] "Z'J"'E F. 

13. Siit Ttt avta — IG: :ipo£ ra A te7)\ inaert iu ras, F. 
16. wpos] eras. V. ^axh F. 17. avxmv^ P; uutiS BFVp; 

om. Anguatna. 18. AS] VE P. FEl A» P. SO, r^s 
AB — 23: sioi^ffOfil «ttl za fj^e p, 21, FE attmov re] 

eras. V, 22, ro .^61] pnnctia notat. F; om. B. S6. 

iexiv B, eras, c 



VM* 



12S ETOrXEUiN 5'. 



"Efftoj o.uota tpiyavtt ra ABF, ^EZ tatjv fj;ovTK 
t^v w()Og Tt5 B yaviKV Ty Jtpos ra E, mq Sl lijv 
AB XQog f^v Br, ovrms r)]v ^JE Jtpos t*jv EZ, 
totfrt o^o^Aoyov dvai, tijv BV tij EZ- Xiya, oti th 
5 ABF tfiiyavov nQog to ^EZ tQiycovov SmXttaiova 
Xoyov Sxsi ^TiBff ri Br apog triv ,EZ. 

EiX^^&a yaQ tmv BF, EZ tQCtr\ KvoiXoyov i\ 
BH, mSts itvtti (og ^^" Br nQog t^v EZ, ovzns 
ti}v EZ jtpcs zi}v BH' xal iaetsvxd-ta rj AH. 

10 'Entl ovv ietiv mg ij AB «pog t^u BF, oOtos 
tl /iE agbg tj]v EZ, ^waAAal apK ietlv ag ij AB 
srpog ti\v /JE, outws ij ^^ tpog rijv EZ. «AA' ws 
^ Br rcpos EZ, oCtojs ^OTiv 17 EZ jtQog BH. xal 
mg aga j; AB jrpos -::/£, ourcis ij £Z ^Tpog Bl/" 

16 rmv ABH, ^dEZ Kpa tQtymvmv dvtiaBa6v&aOi.v at 
nXsvQal ttC jisqI tag tOag ymvlag. tov 8i (iiav fitK 
fSriv ixovtmv ymviav tQiycovmv avrtaiTCov&aOtv a[ 
aXevQal ai xeqI tas tOag ymviag, fffcc iotlv ixetva. 
l'aov aQa fffil 10 ABH TQiymvov rra jJEZ XQt- 

20 ymvm. xal iasC iottv mg ij BF nQog tijv EZ, 
ovTffls 57 EZ tCqos t^v BH, iav S^ tQstg ev- 
^stai ttvdXoyov mGtv, tj jtptoTjj jrpog t^v tQitfjv 8i- 
jtXaOiova Xoyov ixii ^mq ffpog tjjv davtiQav, r^ BT 
aQa aQog T^w BH dtitXaaiovtt Xoyov ix^i ijncp ^ 

26 FB iiQog zrjv EZ. ag 6i 17 FB JtQog tijv BH, 
omag to ABF tQiytavov Jtpos to ABH tQCymvov 



2, tiji B] 10 B V, et F, aed corr. 3. xiiv BT] BFBp^ 

rny F^ F; litt. B in ras. m. 2 V, i^jr EZ] EZ Bp. 8. 

oijito PBp. 10. -4B] B in raa. PF. t^*] om. BFp. 

o5t» P. U. t^v] om. BFp. 12. r^»] bis om. BFp. 

13. «eOB EZ] Bopra m. 2 F; ;ipoe 1^» EZ V. r^* BH V, 



^ 



ELEMENTORUM LIBER VL 129 

Sint similes trianguli ABFj jdEZ angulum ad 
B positum angulo ad E posito aequalem habenteS; 





et ABiBT^ JE: EZ, ita ut BF lateri EZ respon- 
deat. dico, esse ABF : JEZ — BF^ : EZ\ 

sumatur enim rectarum ^J", EZ tertia proportio- 
naUs BH [prop. XI], ita ut sit BF^EZ^EZ: BH, 
et ducatur AH. 

iam quoniam est AB : BF = JE : EZ, permu- 
tando erit AB:JE = Br:EZ [V, 16]. sed BF^EZ 
= EZ : BH, quare AB : JE = EZ : BH. itaque in 
triangulis ABH, jdEZ latera aequales angulos com- 
prehendentia in contraria proportione sunt. trianguli 
autem unum angulum uni aequalem habentes et quo- 
rum latera aequales angulos comprehendentia in con- 
traria proportione sint, aequales sunt [prop. XV], 
itaque A ABH= ^dEZ. et quoniam est BF^EZ 
= EZ: BH, et si tres rectae proportionales sunt, 
prima ad tertiam duplicatam rationem habet quam 
ad secundam [V def. 9], erit BF: BH = FB^ : EZ\ 
sed FB : BH = ABF: ABH [prop. I]. itaque etiam 



14. AB] B eras. F. triv JE Y. triv BH V. 16. &Qa\ 

supra m. 1 p. 17. tQiymvaiv'] om. Theon (BFVp). ,19. 

dEZ] Z paene eias. V. 22. dmlaaiovaova P, sed corr. 

m. rec. 28. ^xv P- ^ ^] ^^ seq- ras. 1 litt. P. 24. 
BH] seq. ras. 1 litt. P. 26. FB] (prius) BF V. 

Euolides, edd. Heiberg et Menge. II. 9 



130 ETOrSEISN 5'. ■ 

xal ro ABF «pa T{f(yavov irpos ro ABH dutXaatova 
loyov Bxn ^jisp tj BF apog t^i/ EZ. i'0ov 3h ro 
ABH rgiyavov rra ^EZ tqiymva' xal rb jiBF 
«p« r^iyavov jrpog ro ^EZ T()i.y(i)i/ov StTcXattCovtt 
6 koyov i'xfi ^KEp ij Br jrpoj r^v £Z, 

T« «pa ofioiu rgiyava jrpog a^^i;Aa ^v SmXaaCovt 
loya iarl xmv o^oi.6yBiv nktVQmv [owfp iSei. ij£C|«:(].J 

TIoQiaiia. j 

'£x (J^ rovrou qooivFpov, oit, iav rpets Evd^crai 

10 avttXoyov (oaiv, ioziv las ii itfiiatij npog rijv rffixriv, 

ovT tag tb a^b r% wpojrjjs f^ffos rapog to anp t^$ 

Sivrigastb ofiotov xal otioicog dvaypaipofitvov [ixtijteQ 

i6ei%&ri, ag ^ FB Jipog BH, ovta>s rb ABF rgiyavov 

xgbg ro ABH r^iyavov, tomiert ro z/£Z]. ontff . 

16 ftfet dti^tti.. 



Ttt ofioia xoXvyava tig te o^ota tgiyiav 

6t.ai,Q£tTai xal Eig laa ro ;tA^9os xk2 bfioXoya 

rotg oXoig, xaX rh xolvytavov srpog to xoXv- 

ytovov SmXaaiova Xoyov f'j;£t ^irip ^ bftoXoyos \ 

«XtvQtt Jrpoff T^v bfioloyov nXtvQdv. 

"Eflrra o;iO(K jioAtJyePva ra ABT^E, ZH&KA,M 
o^oAo/os tfi i^uria ij ^S r^ Z/f' Kiyo), ott tu ABr^JE^ T 



XX coroll. EutocinB in Arcbim. UI p. 6 



I. opa] om. P. AEH] B Bnpw m. 2 in ras. V. 

ioiiv BP. 9. tB»] i- in ras. m, 2 V. 10. soiti'] om. Bp. 
11. BtSog] P; Tefyinvo» Theon (BFVp), comp. aupra P i" 
rec. 13, i^» BH V. 14. TO] om. V. lODttW P. ti 
eupra m. 2 F. 16. iti^ai] soi-^aai V. 19. Siloif] pOBt o 
1 litt. eras. p. 30. ij] om. B. 22. ABTJE] ABr^EZ,M 

P, eed. corr. ■ 



I 

I 



ELEMENTORUM LIBER VI. 131 

j4Br: ABH= Bn : EZ^. erat autem ABH= ^EZ. 
quare etiam ABT: JEZ = BI^ : EZ\ 

Ergo similes trianguli inter se duplicatam rationem 
habent quam latera correspondentia. 

Corollarium. 

Hinc manifestum est, si tres rectae proportionales 
sint^ esse ut prima ad tertiam, ita figuram in prima 
descriptam ad figuram in secunda similem et similiter 
descriptam.^) — quod erat demonstrandum. 

XX. 

Similia polygona in triangulos et similes et aequales 
numero et totis correspondentes diuiduntur; et poly- 
gonum ad polygonum duplicatam rationem habet 
quam latus correspondens ad latus correspondens. 

Sint similia polygona ABT^E, ZH&KA, et AB 
lateri ZH respondeat. dico, polygona ABF^Ey 

1) Hoc ex proportione ABF: JEZ = BFz BH concludi 
noluit Euclides, paullo audacius sane; nam huic corollario 
post prop. 20 demum locus erat. sed xqlyatvov lin. 11 sine 
dubio Theoni soli debetur; nam Bldoq tuentur P et Gampanus 
et aliquatenus saltem Philoponus et Psellus (hic corollarium 
8U0 numero citat) xBxqdyoivov praebentes, quod cum scriptura 
Bl6og conciliari potest, cum xqCytovov non potest. et prop. 20 
coroll. 2 in P in mg. additum et a Campano omissum a Theone 
interpolatum merito uideri potest, id quod et ipsum sen- 
tentiam meam de huius corollarii forma confirmat. tum 
Pappus VIII p. 1100, 15 nostrum locum respicere putandus 
est, et sane scriptura eius loci tam incerta est, ut inde do^ 
numero, quem indicat, corollarii nihil adfirmari possit. itaque 
puto, Euclidem ipsum Bl6og scripsisse et Theonem, quo corol- 
larium facilius pateret, nostrum locum mutasse et prop. 20 
coroll. 2 addidisse. sed uerba inslnsg lin. 12 — JEZ Im. 14 
interpolata esse putauerim, neque Gampanus ea habuit; sed 
Theone antiquiora sunt. 

9* 



132 LTOIXEIiiN 5'. 

ZH&KA TtotvYOva sig ts o^ota zgiytava Siaipsttt 
xal tig lea z6 iilij&os xal ofioloya toI$ oloig, xal 
To ABFiiE «okvytovov tcqoq to ZH&KA aokvyDJVov 
SinkaeCovR loyov £xh ^tibq ij AB npos ti;v ZH, 
5 'Exsttvx^taaav at BE, EF, HA, A&. 

Kal insl ofiotov iati to ABTAE zokvyav 
ta ZH&KA nolvy^va, terj iatlv •tj vno BAE ym- 
via tij vno HZA. xaC iotiv atg ^J BA jrpog AE, 
ovrag rj HZ wpoff ZA. ixtl ovv dvo tgCyavii iatt 

10 zic ABE, ZHA fiCav yaviav (iia yavCa tetjv ixovva^ 
stepl &£ Tils teas ymvias rag nlEvgas avaloyov, 
ieoymviov apa iatl ro ABE tpCyatvov tp ZHA 
vffiyavD)' metB xal ofiotov fffjj ap« ievlv ^ vao ABE 
yeavia t^ vno ZHA. iiJrt i\ xal oXt] t] vko ABF 

15 oAjj r^ uffo ZH& teij dta ttjv ofioiottjta ttBv JtoAu- 
yiavtoV Aocn^ aptt j/ vno EBF yavCa TJj vno AH0 
ietiv iat]. xal iml Sia ti]v 6f(oidrJjra Trow ABE, 
ZHA tQiydviav iariv tog i] EB jtQog BA, ovrat? 
5j AH n^og HZ, «AAk fitji' xal Sia t7]v 6iiot6tr]tt 

20 TOi' nolvytovav ietiv tag i] AB -xqos BF, 

ZH JTpog H&, di' teov aQa ieilv tog rj EB xqos 
Br, ovtas 1) AH nQog H&, xal nspl tag teaq yca- 
vCas Tag vno EBF, AH& at jtKBVQoi avakoyov sleiv 
iaoytnviov uQa ietl lo EBF tQCyatvov ta AH& 

26 Tffiymva' Sots xal ofioiov iatt to EBV TgCymvov 
Ttp AH& tQiydva. Sia ta avta Si] xal t6 EFJ 
TQiyavov Sfioidv ieti t« A&K TQtytova. 

6. AS} nvutat, in JE F. 7. inl aeq. rag. 8 litt i 

6. HZA'] ZHA F. Ttiv AE V. 9. HZ] ZH P. 

ZA V. 10. ytavitt] ytavCav Yq?. 11. 34] om. F. 

fin;] oorr. ei taov m'. reo. P. IB. ZH8] H nidetnr corr. "V 



'.ttat ^^ 

A 

'V 

I 



m 



ELEMENTORUM LIBER VI. 



Ij^ ZHQKA in triangulos et simi- 

,' ""^,. les et aequales numero et totie 

'VT^ff /i % cofrespondentes diuidi, et esse 

\\/ S^^y* ^BTJE : ZH®KA = AR* 
r J « K AMH^cakxxt BE,Er,HA,A®, 

etqiioniam ABTdE~ZH®KA, er\tLB^E=HZA 
[def.l]. etBA-.AE-^HZiZyllid.]. iamquoniamdno 
trianguli suut ABE, ZHA unum angulum uni angulo 
aequalem habentes et latera aequalea angulos compre- 
liendentia proportionalia, erit A ABE triangalo ZHA 
aequiangulus [prop. VI], quare etiam aimiles sunt 
[prop.IV; def.l]. itaque/.-JB£= Z//yJ. uerum etiam 
i ABr= Z H@ propter similitudinem polygonomm. ita- 
(iaeLEBr= AH®. et quoniam propter similitudinem 
triangu!orum^B£, ZHA est EB:BA = AHiHZ, et 
praeterea propter similitudinem polygonorum AB:Br 
— ZH : H®, es aequo erit EB i BF = AH:H& 
[V, 22], et latera aequales angulos EBF, AH® com- 
prehendentia proportionalia sunt; itaque A EBFtrian- 
gulo AH@ aequiangulus eat [prop. VI]. quare 
C^EBFr^ AH® [prop. IV; def. 1]. eadem de causa 
^M etiam A ETj^ -^ A®K. itaque similia polygona 



1 



\ 



16. !§] P, P m, 1; XoiTti t^ BVp, F m. 2, 17. Snj 

iax{v F. 18. xriv SA V. 19. AH^ ABtp. tfiv HZ V. 

20. zriv SrV. 21. ZH] HZ P. rt^v HBV. He,it 
r«oti] ip; uidetur fnisse alia scnptnra a m. 1. EB] £ e 
corr. F. 22. 117* BT V. Tt,v H@ V. 23. */oi»] om. V. 

2t AHB] ,10HV. 26. Ioti] om. BVp. li EBT-- 26: 
■tfiymvui] mff, m. 2 V; F haec uerba ut cett. codd. in t«ita 
habet, sed dein io mg. m. 1 : mett xkI o^oiDf ro EBF xA 
AUe tfiyav^. 27. ASK} ABHtp; corr. es AKQ m. 1 p. 



J 



134 ETOIXEIHN s . 

o/ioca aokvyava t« ^BF^E, ZH&KjI eCs te oftoi 
TQiyavtt d(^pjjt«[ Kai. eig laa to JtA^fl^og, 

Aiyat, Ztt xal ofto^oya toig oXoig, tovziozu 
aOtB aualoyov etvai ta tgiyava, xal tiyovfisva [i'i 

5 eivai T« ABE, EBF. EFJ, inofiBVK di avtmv 
ZHA, AH&, A&K, xal ozi zo ABFAE noXvyavi 
7ti»og ro ZH&KA aoXvyavov dixlaaiova ).6yov i^' 
^Kfp ij oiioXoyos iilBv^a Kpog tijv OfioXoyov nXBVQiiv^ 
tovziaziv 1} AB JTpos tijv ZH. 

3 'EaB^evx&aOav yuQ al AF, Z&. xai ixtl 6ia ti/v 
ofioiotrita tiav noP.vytovmv fffjj ietlv ^ u/ro ABF yiovCa 
Tt} V7c6 ZH&, xa( iati.v ag ^ AB TiQogBF, ovzag i} ZH 
}t(fos H&, ieoydviov iati lo ABF rQLyavov ta ZH& 
TQiyava' [a>j aga iatlv ij (ihv vno BAF yavia TJj vito 

B HZ&, ij 61 vno BFA Tij vno H&Z. Koi iael Hdr} iativ 
ij vjio BAM yavia ty vjto HZN, iati 6% xal rj into 
ABM tfi vnb ZHN teri, xal Xoinii aga ij vno AMB 
}.ont^ zfi VTtb ZNH l'ai} ietiv laoymvtov epa iaxl 
to ABM TQtyavov rp ZHN tQiyava. ofioias 8ij 

dEi^ojiBV, oTt xal ro BMF tpiymvov iaoydvtov iott 
1(5 HN& TQiydva. avaXoyov rep« ietiv, og (liw 
AM jtgog MB, ovrag 17 ZN irpos NH, tos di 
BM Jtpog Mr, ovTiog fj HN ngog N&' toatB x 
Si' taov, ag t; AM nQog MF, ovrras 1; ZN Bpoj 



I 



2. diBietftai qj- els] om. BV. 5. ABE] E in ras. P. 

avTWv] aic tp, eed oBtore F. 6. A9K] &KA F. 3w] 
-i in raa. P. 7. jtoluyeoto»] -vov BUstulit litcana pergam., 

Bnpra sor. rm in. 2 F. la. i^» BT BFVp. 13. ijj» H» V. 
1<IT-] aoa iatiF. 14. fflij] -tj in raa. P. B^rl ABT F. 

15. HZel H corr. es Z p; ZH0 F. JfSZ] eHZ F. 

16. BAM] PVp, B m, 1; "A'BMF: ABMB m, rec. HZJVJ 
ZHJV in ras. m. 2 B. tati] P; iae{x9ti Tbeon (BFVp). 



J 



ELEMENTOEUM LIBEK VL 135 



W ABF^E, ZH@KA m triaugulos et similea et aeqaalea 
H aumero diuisa sunt. 

■ dico, eos etiam totia correapondere, h. e. ita \it 
trianguli proportionalea sint et praecedentea ABE, 
EBF, EF^ et eorum termini aequentes^) ZHA, AH@, 

»ASK, et praeterea polygona rationem duplicatam 
babere quam latera correspoudeutia, h, e, esae 
ABF^E : ZH&KA = AB^ : ZH^. 
ducantur enini AF, Z@. et quoniam propter 
eimilitudinem poiygonorum est /. ABF = ZH&, et 
AB : Br= ZH: i/®, erit A ^Sr aequiangulus trian- 
gulo ZH@ [prop. VI]. itaque L BAF = HZ® et 
L BFA = H@Z. et quoniam L BAM = HZN et 
L ABM = ZHN [p, 132, 13], erit etiam L AMB 
= ZNH [\, 32]: quare A ABM aequiangulus eat 
triangulo ZHN. similiter demonstrabimus, etiam 
A SMr aequianguium esse triangulo HN@. itaque 

»AM:MB = ZN: NH, BM: Mr=HN:N&[]}iop.lY]. 
quare etiam es aequo AM: Mr = ZN : N& [V, 22], 

1) In avtav lin. 5 uonnihil ofiensionia eat; aed cum fnd- 
fiGVtt idem Bit ac opoi fxdfitvot, genetiuaa feni poteBt. et 
additum uidetur uocabulnm, ut aignifioetur, ZHA esse t«r- 
minnni aeqaeatem trianguli ABE, AHB autem trianguli 
EBr, ABK aatem trianguU ETJ. ceterum commemorandom 
eat, tum demum adparere, triajigaloB totia (.h. e. polygonia 
AEPJE, ZH&KA) cotreapondere , cum demouatratum erit, 
eaaa ABFJE : ZHeKA =• AB^ : ZH\ h. e. — ABE : ZHA 
_ ^EBr-. AHe — ETJ : A9K. 

^^ 17, ABM] mutat, in BAM m. 2 B, ZHN] mntftt. in 

^K^KZiV m. 2 B. AMB] A BM puuctia supra ^ et M deletU F. 
^ ^ 80. itTiv F. 21. 1] ah p, 22. AM] M corr. en S m. 

■ S T, r^iv MB T. NH] N in nu. m. 2 Y. 23. ovitot 



136 STOIXEliiN <;'. 

N@. uXX' ms n AM %qqs MF, ovxaq zo ABSi 
{xglyavov] xqos x6 MBF, xal tro AME Jipos ■ 
EMF' TtQos KXKriKa yti^ sieiv ag al ^deeig. xal t 
aQcc ¥v xav ijyovniviov nQos iV Ttov i%6fisvav, ovri 
SnavxK TN Tiyov^sva xgog aaavra xa iaoiieva' t 
aga zo AMB XQiyaivov wpos ro BMr, ovTtos iri 
ABE npos t6 rBE. dXX' as xb AMB jrpog i 
BMF, ovxag ^ AM ^rpog MF' xal rag «pa ^ AM %(fh 
MFf ovrwg x6 ABE T(/iytovov jrpog t6 EBP tpiyavot 

10 Siit xa avxa Sii xal tog rj ZNngos N@, ovxcos t6 ZH4 
Tpiyavov ZQog ro HA® xpiyavov. xcei iaxiv (og ^ AA 
npog Mr, ovztog rj ZN nQog N&' xal (og Spa to ABi 
zpiyavov ZQog x6 BEF xpiyavov, oikas z6 ZHj 
XQiyavov jtgog t6 HA@ tpiyavov, xctl ivttiKa^ < 
to ABE Tgiyavoy ng^s x6 ZHA TQiyavov, ovtmi 
To BEF tgiyavov jrpog 16 HA® xgiyavov. ojtoiia 
Sii Sti^o^isv iat^svx&sieav tav BA, HK, 01 
(0? t6 BEF rgiyavov Jrpos ro AH® xgiycovoi 
ovtcos t6 EFA tgiyavov xgog to A@K XQiyavot 

80 xttl imi ioxtv as to ABE xgiyavov Jipos t6 ZHj 
tffiytovov, ovxag x6 EBF nrpog to AH®, xal ttt 1 
BFA Jtgos x6 A@K, xccl ag efpa ¥v tav f/yovfiBvm 
jrpos ^v tav iTtonsvav, ovxios amavxa xa riyovnm 
XQog ajictvxa xa inofievtf isxiv itQa tbg x6 ABi 

2b zQiymvov Jrpos t6 ZHAzgiyavov, ovxas z6 ABFjA^ 
nolvyavov jrpog t6 ZH@KA xolvyavov. aila 1 
ABE xgiyavov irpos t6 ZHA tgiyavov Smkaaiovi 
Koyov l%Ei ^itEQ ri AB ofio^oyos nXfVQU np6s x^H 
ZH oftoloyov x^ivQttv' ta yag OfiOta tQiyava . 



ELEMENTORUM UBER VI. 137 

sed [prop. 1] j^M : MF = jiBM : MBF -» AME 
: EMF] naiu eandem inter se rationem habent quam 
baseSv itaqne etiam ut unus terminorum praecedentium 
ad unum sequentium^ ita omnes praecedentes ad 
omnes sequentes [V, 12]. itaque AMB iBMF^^ABE 
: TB E. sed AMB : B MF —AM: MF. quare etiam . 
AM : MF = ABE : EBF. eadem de causa erit etiam 
ZN : N0 — ZHA : HA@. et AM : MF^ ZN: NS. 
quare etiam ABE : BEF = ZHA : HA&^ et permu- 
tando [V, 16] ABE : ZHA = BEF : HA@. similiter 
demonstrabimus ductis BjJy HK, esse BEF: AH& 
— Er^ : A@K. et quoniam ^t ABE:ZHA = EBT 
: AH@ = EFjJ : A@Ky erit etiam, ut unus termi- 
norum praecedentium ad unum sequentium, ita omnes 
praecedentes ad omnes sequentes [V, 12]. itaque 
ABE:ZHA = ABrJE:ZH@KA. sed ABE:ZHA 
= AB^ : ZW] nam similes trianguli duplicatam inter 



to^om. P. 4. &Qtt] om. V. 8. ti^v MF V. 9. triv 
Mr-Y. 10. NS] N in ras. B; HG tp (non F); njv N9 V. 

11. To] om. P. 12. t^v Mr BFVp. triv NS FV. 14. 
HAG] corr. ex HQA m. 2 V. 16. BEr] EBT V. ff^©] 
mutat. in AH@ m. 2 V. 18. BET] P, V m. 1; EJBTBFp, 
Vm. 2. 19. ETJ tQlyavov] P; ETJ Theon? (BFVA 

20. xal 1««^ iativ cogl mg. m. rec. P. ^ 26. ZH^] 'H"ZA¥. 
Post ovTCDff eras. Trpds Y. 29. yctQ] &Qa qp. 



138 



ETOIXEI-QN ?'. 



dmXeeaiovi Xoyo) iari tcov ofioXoyav jilevQiBv. xal ri 
ABT^E ttQK jToAuytovov repos to ZH&K.d xoXvya- 
vov fimXttOiova koyov ^x^t ijnsQ 7} AB oftoAo^s 
nXBVQK TtQbs zriv ZH 6fi6i,oyov TcXsvQav. 
B Ta Kpft of^otK aoXvyava BCg te ofioitt zgiyavtt 
dittiQstzai xal fCg taa zo xk^&oq xal b^oKoya rofs 
okots, nal zb nolvyavov npbg zb no?.iytavov dmla- 
eCova koyov Ix^i ijxEQ ^ ofioAoyog mXtvQtt «pog z^v 
biioXoyov nXsvffdv [oBcp ISst Set^ai]. 



IIoQiSfia 






'Heavzcag d% xai inl zmv [ofioiaiv] ZEiQaaXEvqav 
iSBi.%&'q0sztti., ozt iv dtitlaeiovi loy^ bM tmv bfio- 
Xoymv nXsvQatv. idtix^V <^* '^**^ ^'"-^ ^*^'" tQtyinvmv 
a6t£ xal xtt&oXov za ofiota £v&vypa(i(ia Ox^^fiBa 
15 npos ttXXrjXa iv diaXaaiovt Xoya tlal zav 6(toX\ 
jtXevgiav. onEQ aSei, det^at. 

[noQtefia j3'. 

Kal iav itov AB, ZH tQLXrjv avttXoyov Xdfiei- 

Hev tijv 3, i] BA itQog Ttjv 3 SiJtXaOiova Xoyov 

20 ix^t ijiteQ 7j AB nQbg tijv ZH. i^jjEt Sl xal ro 
aoXvytovov nQog ro xolvyavov ^ to zezQanXtvQov 
jrpog ro zeTQajtXevQov SiaXaaiova Xoyov ^jtep rj b(i6- 
Xoyog nXevQtt JtQog t^v bfioXoyov nXsvQav, zovziuriv 
17 AB JtQog zi}v ZH' iSeix&ii &b zovzo xal inl ztov 

S6 ZQtyavcav' {afSZB xal XKi&oADU (pavBQov, ori, iav Tpe^, 
Bv&Btai ttVttXoyov aSiv, Sazttt aig 57 aQiofri irpos 
ZQiztjv, ovtmg zb dab z^g «poiTjjg Eitfog itQog zb 
zTig SBVTEQag zb Oftotov Kal bfioiag dvayQaipbfi^BvovJ] 

1. iaxiv F. 2. jiolvycirov] (alt.) ^ioXvyovov p. 7. jroiliiyo)- 
»ov] (alt,) jtolvydvtov ip. 10. xopiiTfta] om. PBV; Na' Fp, 11. 



i 






ELEMENTORUM LIBER VI. 139 

86 rationem habent quam latera correspondentia 
[prop. XIX]. quare etiam 

ABFJE : ZH@KA = AB^ : ZH^ 

Ergo similia polygona in triangulos et similes et 

aequales numero et totis correspondentes diuiduntur, 

et polygonum ad polygonum duplicatam rationem 

liabet quam latus correspondens ad latus correspondens. 

Corollarium. 

Et similiter etiam in quadrilateris demonstrabitur^ 
ea duplicatain rationem habere quam latera correspon- 
dentia; et idem in triangulis demonstratum est. quare 
omnino similes figurae rectilineae inter se duplicatam 
rationem habent quam latera correspondentia. — quod 
erat demonstrandum. 



cocravTQ)?] C9- m. 2 V. ofio^ov] supra m. rec. P. 12. bIcCv F, 
katl Bp. 16. Blci] PV, P m. 2, p; bIciv B; hxi P m. 1. 

16. o«fp i8Bi dffgat] P; om. Theon (BFVp). Totum co- 
xollarium om. Campanus. 17. noQiCfia §'"] om. codd., seq. 
cum coroU. priore coniunctis. lin. 18 — 28 in mg. inferiore m. 1 P 
pro scholio, signo ^^ hnc relatum. 18. ZH] H in ras. F. 

19. f^v IS}] seq. ras. 1 litt. V; corr. ex r^i NS P. "n ^^] 
e corr. F. S] post ras. F, ante ras. V (1 litt.). 20. AB] 
BA F. 21. ^J corr. ex xat m. 2 V; om. Bp. 23. TtXev' 
Qocv] P, om. BF Vp. 25. noQvofia mg. BVp. xal qpaveodv p. 

27. sl6os] sequente ras. 1 Utt. qp (uestigia sunt syllabae 
'Ov F). nQog] supra V. 28. Sequitur alia demonstratio 
secundae partis propositionis , qusie u. in appendice. 




ETOEEIiiN ET'. 



Ttt ra ttVTtp £v9v'yQc<iiiia Siioiix xal hAAjj- 
Xoig ierlv ofioitt. 

"Etfrca yaff ixaTSQOv t<dv ^, B sv&vyQiifificav i 

5 r ofioiov kiya, ori xal t6 A ra B ieriv oiioiov.^ 

'Entl yaQ ofioiov ieti t6 A rc5 F, tfSoyfoviov 

ri iettv avra xal Tag xipl iKg taas yaviag nXtV- 

pag ttvaioyov ix^t. nuXiv, insl oftoiov ieri lo B 

rp r, laoydviov ts ieriv avra xal rag itepl Taq 

io i'aag ycavCag nXBVQaq avdXoyov ix^i. ixaTEQOv apu 

riav A, B r<p P (soymviov ri fUn xal rag izeqI ras 

teag ytaviag xltVQag avaXoyov ix^i [oIctc xkI i 

rp B laoytaviov ri ian xal rag mpl tks fffoff ] 

viag itXEVQttg avciXoyov Ix^i]. ffftotov aQU ieil tol 

15 Tci S" oTciQ ISsi dEiiai.. 



'Eav xieeaQsg BV^Btat, avdkoyov foetv, i 
tu Kx' RvTfov iv&vyQa(i(ia ofioid te xal ofioiS 
uvttyEyQa(i{iivtt dvdXoyov larai,' xav ra ^S 
3 avTiBv sv&vyQRfifia ofioid te xal o(to/taff dva- 
yeyQafifiivtt ttvciXoyov r;, xal avtal aC evifffBt 
KvdXoyov iOovrai. ■ 

"EaTcoeav riaaaQEs EvQeiDi. dvdkoyov ai AB, ^^| 



1. %a"\ m. 2 V; «y' Fp. 4. 
corr, m. 1. 6. httv ouotov V. 
jtaXiv iietll in raa. m. 2 P. iattv qj, 

11. Tf] om. V. 12. laasl Bapra m. 

A — U: avdXoyov ^iu\ Tteon? (BFVp)! 



Deinde x^^^opoHitiont 



. repetit AngnBtus, ut fieri solet. 





ELEMENTORUM LIBER VL 141 

XXL^) 

Quae eidem figurae rectilineae similes sunt figurae, 
etiam inter se similes sunt. 

^^.^^^ Sit enim utraque figura rectilinea 

\ ^^ V^ Ay B figurae F similis. dico, etiam 

\ ^l \ — i figuras Ay B similes esse. 

nam quoniam A figurae F similis 
est^ et aequiangula est ei^ et latera 
aequales angulos comprehendentia pro- 
portionalia habent [def. 1]. rursus quo- 
niam B figurae F similis est^ et aequiangula est ei^ et 
latera aequales angulos comprehendentia proportionalia 
habent [def. 1]. itaque utraque figura A^ B et aequi- 
angula est figurae Fj et latera aequales angulos com- 
prehendentia proportionalia habent. quare A f^ B 
[def. 1]; quod erat demonstrandum. 

XXIL 

Si quattuor rectae proportionales sunt^ etiam 
figurae rectilineae in iis similes et similiter descriptae 
proportionales erunt; et si figurae rectilineae in iis 
similes et similiter descriptae proportionales sunt, 
etiam ipsae rectae proportionales erunt. 

Sint quattuor rectae proportionales ABy F^, EZ, 



1) Nam corolL 2 p. 138, 17—28 Theoni uidetur deberi; u. 
p. 131 not. 1; om. Campanus (sed is quidem etiam coroll. 1 
omisit), et in B adscribitur mg. m. rec. iv &Xlq} ov yQatpBtai 

TOVTO. 



xj3'] %$' p et F, sed corr. m. rec. 17. mciv] P et B, sed v 
eras.; mai FVp. 23. sv&sta F. 



142 



STODtEIGN ?'. 



.ojti^ 



AB Jtpog rijv r^, otJtos ^ £Z 
ffpog rijv H&, xal avaysyptt<p9aeuv «no fiev Tmi' 
^S, r^ o|[iota tE xal oiioiag xBLfiSva Bv&vyQaii[ia 
1« KAB,Arj, uTtb 6% rav EZ,H® onoid ts xaX 
5 oftotos xeC^Eva ev&vypaiifia t« iWZ, A'®' Xiyta, Sn 
^Orix' to^ ro KAB jrpog lo AFjA, oikras ro JtfZ 
n:pos To iV®. 

EiX^^fp&co yuQ xav [Iev AB,'r^ Tptrjj avdh 
f/ S, Tmv dh EZ, H@ tptri) dvdXoyov ^ O. 

10 ^wf^ fBTtv atg filv i] AB tiqos t^v F^J, ouTOg ij 
£Z wpog rijv if®, (»5 ^^ h r^ ^Qog r^J/ S, ovzag 
ij H® KQog T^v O, SC feov uqu tgzlv ag ^ AB 
^Qog zijv !S, ovzag ^ EZ nQog z-^v O. alk' ag ft\v 
7) AB ngbg rijv IS, owrog [xal] ro KAB wpos to ATA^ 

15 ag Ss ^ EZ TtQog zrjv O, ouTog ro MZ Ttffbg zb 
N®' xal ag apa ro KAB Jipog ro AFA, oi^Tmg 
MZ ;rp6s ro N@. 

'AKi.a Sij ^'ffrtj tog rt KAB jrpog ro AFjd^ 
zag zb MZ JTpog ro N@' Xiya, ozi iezt xal ag 

20 AB Wpog T^v FA, ovvag rj EZ repog tr]v H@. 
yaQ ftij iaziv, tog f] A3 itQog z^v FA, ovtas 
EZ mpog zi}v H@, Sata tog h -^^ Jtpog r^v fj 
ovtag ij EZ repos zijv HP, xal dvayayQceq>&a a: 
T^S HP oxozBQa raiv JVfZ, N@ o[ioi6v ze xttl ofio^o? 

■15 xdififvov tv&-vyQtt(iiiov tb ZP. 

^Eitel ovv ietiv tDS ^ -^B ffpos rijv ri/, oi^tiog 

1. AB] B fiupra m. 1 P; postea insert. F. EZ] in rai. 
m. 2 V; ZE Fp. 2. «yoyt7p«ipni(i«v p. 6. JtfZ] Z e 

corr. F. Post Sti tas. 2 litt. F. 6. AFJ] Utt. AF in ras. ra. 
2 V. 11. r^l .d etas. V. 13. EZl e corr, Ytp. 14. «oi'] 
om. P. jr-J] litt. ^r in ras. m. 2 V, r^-ijp. 16. '" 
cf«a — 17: 10 IV»] om, BVp. 18. AFjJ] FJJ q>. 



lul i»( 





ELEMENTORUM LIBER VL 143 

He, ita ut sit ABiFJ^^EZiHS, 

et in ABj FjJ similes et similiter 

positae figorae rectilineae descri- 

^ ^ J bantur KAB, AFJ, in EZ^ HS 

•^j—l autem similes et similiter positae 

j[ (9 figurae rectilineae MZ^ NS. dico, 

esse KAB : AFJ — MZ : NS. 

Sumatur enim rectarum AB, FA tertia propor- 

tionalis S^ rectarum autem EZj HS tertia 

S ' — ' proportionalis O [prop. XI]. et quoniam est 

E AB:rA = EZ:HS et rA:S = He:0^\ 

JL O ex Aeqxio eni[Y,22]AB:S = EZ:0. sed 

^-^ AB:S^ KAB : ATA [prop. XIX coroU.] et 

EZ:0 = MZ : NO [id.]. itaque etiam 

KAB : AFA = MZ : NO. 
Uerum sit KAB : AFA = MZ : NS. dico, esse 
etiam AB : FA = EZ : HS. nam si non est 

AB : TA = EZ : HG, sit AB:rA = EZ: UP 
[prop. Xn], et in TIP utrique MZ, NS similis ef 
similiter posita construatur figura rectilinea 2P 
[prop. XVin et XXI]. 

lam quoniam est AB : FA = EZ : UPy et in AB^ 



1) Nam ex hypothesi est AB i TJ ^ TJ i ^ eiEZ i HB 
= HB : O; et AB i TJ =- EZ i HG. 



Ard] TAJ F. 19. To] (prius) eras. F.^ inxiv PJB; comp. p. 
20. H yctq iiT^ iaxvv, ms 7^ AB tcqos ttv Fd, ovtas rj EZ 
TCQOs^ tfiv HS] mg. m. 1 P; om. Theon (jBFVp). 22. iatm 
ms Ti AB TtQos TTjv Fd, ovTOff ri EZ nQOs t^v IIP %al dva- 
yfy^aqp^o)] P; ysyovstoi} yuQ (og %tX, Theon (BFVp), P mg. 
m. rec. 23. dvaysyQatpo* p. 24. OTEors^a qp (non F). 25. 
av^vygafifiov] om. BFp. 






144 ETOIXEiaN 5', 

7i EZ nQog TijV TIP, xal avaysyQanTai axo (liv 
tmv AB, r^ ofiaia rt xal onoiag xeifitva t« KAB, 
AF/}, ano ds rmv EZ. ITP ofioia w xal 6(ioia>g xei- 
p.Bva tff MZ, EP, t6riv Sga ois to KAB 3rpQ$ th 
5 AF^, ovreas to MZ jrpog ro £P. vnoxnTai di xat 
mq To KAB JTpog t6 AF^, ovrag ro MZ itffos to 
N&' xal mg «qb ro MZ «pos ro 2^P, ovros to MZ 
^Qog To N&. To MZ «p« npos ixarsQov Tav N@, 
SP Tov avtov i%H loyoV t^ov a^a iarl rb N& ip 

10 2P. ^ett. di avzm xal ofioiov xal 6,uo^(»s xeifievov 
/■ffjj apa ij H@ Tfj 77P. xal ind iativ dtg 17 AB 
wpog t^w r^, ovTias ij EZ 3tpos rijv UP, Ioj} Sh 9. 
TIP xf) H&, iOTLv UQa atg ij AB srpos r^i' fv^, 
TWS ^ EZ JTpOg T1JV if®. 

iB 'Bav «pK rdaaaQEg sv&eiai 'avakoyov loeiv. xal 
ta (Jjt' avTmv svd-vyQafijia ofiota ts xal bfioCatg dva- 
yeyQafifiiva avdloyov iatat- xav ra ajr' avr^v ev- 
^vyQUfifia ofioia te xa\ bfiolcag avayByQafiftiva ava- 
loyov t}, xal avtal aC svd^etai avdloyov iaovtttf 

30 oiT£p idsi dst^ai. 

[Aijfifia.] J 

["Oti de, iav svd^vyQafifia i'aa jj xal ojitoia, 4 
bft6f.oyoi avtmv }ci.£VQal taai dXk^laig slelv, Ssl\opKlt 
ovtag. 

25 "Eatat lea xal ofioia si&vyQafifia ta N&, £P, 
xal Satm <ag ij @H Jtpog rijv HN, oijtmg ij PJl nfbg 
* tiiv n£- kiya, oti taij ictlv i PII tH @H. 

Bl yiiQ KVieoi siaiv, fiia avtmi- fisi^mv iativ. 

2. KAB, ATJ] B, AP litt. in rat 
JTP duae litt. del. m. rec. P. 7. JVS] 




ELEMENTORUM LEBER VI. 145 

r^^ similes et similiter positae' descriptae snnt KAB, 
AFjdy in EZy TIP autem similes et similiter positae 
MZ, EP, erit KAS : AFA ^ MZxHP [u. supra]. 
86d suppOBuimus, esse etiam KABiAFA^MZiNS. 
itaque MZ : EP « MZ : NS. itaqpe MZ ad utram- 
que NS, SP eandem rationem habet. quare NS = SP 
[V, 9]. uerum etiam ei similis est et similiter posita. 
itaque HS = 11 P}) et quoniam est AB:rA = EZ: IIP, 
et nP=H@, erit AB:rA = EZ : H@. 

Ei^o si quattuor rectae proportionales sunt, etiam 
figurae rectilineae in iis similes et similiter descriptae 
proportionales erunt; et si figurae rectilineae in iis 
similes et similiter descriptae proportionales sunt, etiam 
ipsae rectae proportionales erunt; quod erat demon- 
strandum. 



1) Nam cum NS : ZP =- H9^ : ITP* [prop. 20] et 
NS — 2;P, erit JTP* =- HO»; h. e. /IP -« HB, 
et hoc ipsum nia indirecta in lemmate ostenditur; sed com a 
ratione Eudidis abhorreat, eius modi res postea demum demon- 
strare nec suo loco in demonstratione insertas, puto, lemma 
subditiuum esse (sed Theone antiquius est); om. Campanus, 
nec res propria demonstratione eget. 



corr. ex £PP, in ras. Y; supra hoc uocabulum et proxime 
sequentia in V ras. est. MZ] in ras. V; Z insert. m. 1 F. 
8. NB"] in ras. V. 9. X^yav i%u p. ioxCv P, comp. p. 

10. CLvxo p. 11. a^a] supra add. %ai m. 2 comp. F; a^a 

ictiv y. 15. dai Y. 16. avaysyQafifiiva^ seq. insert. m 
ras. m. 1 F. 18. xat] m. 2 V. 21. Xijfiua] xs' p et e 

eraso F; m. rec. PBV. 22. Si] m. rec. P.' ^l om. V. 

Post ofioia add. V m. 2: iavtv. 23. siai BFYp, iiiiofi.Bv 
corr. ex 9ei%<ofi.6v m. 1 P. 25. xd] e corr. V. NS, £P 
inter iV et ras. 1 litt., item inter 2; et P V. 26. PJT 

mutat. in JTP m. 2 V; ITP Bp. 27. ti^v] om. F. 28'. 

aviaog V. Btctv] PB; Blai Fp; iaviv V. 

Enolidef, edd. Heiberg et Menge. II. 10 



146 STOIXEISN S'. 

ioro) fiti^ayi' rj PTI r^j &H. xal insi iOTiv dts 1} 
PII it^hg n£, ovTcos ^ &H rrpis rijv HN, xal 
ivaXla^, rog ij PII npos rijv &H, ovrras ■^ n2J jrpos 
i^v HN, (isitt^v dh ij UP z^s ®H, }iE{%<av apa 
xcti ij nS T^s HN- aatt xal z6 PZ fiettov iGri, xov 
&N. aXlit xal ieov OKEp advvarov. ovx apa avia6s 
ifSrtv rj nP T^ H&' (ijij «p«' onrfp tSci Set^ai,] ^^^ 



xy. 

Tu ieoydvta naQaf.iTjl6yQafiiia XQtis K^Aijila 

10 Koyov l%Bt zov evyxiifisvov ix tav nXBvffcov. 

"Eeta leoydvia na^aXXijX6yQaii[ia ta AV, VZ fffijv 

£%ovza triv vnb BF^ yaviav zij vno EFH' Xsyat, 

Oti 10 j4r 3iapaXXrji.oyQaii[iOv irpos zo VZ xa^aXXi]- 

X6yQa[i(iov Xoyov l%ct zov evyxtijisvov ix zav nXsvpmv. 

5 Ktie^a yaq aete in' ev&fias ilvat rijv BV TJ) 

FH- in' Ev&aias aga iorl xal rj ^P r^ FE. xal 

ev(i7tEJtlriQde&e3 to z/H naQaXXrjXoyQafiiiov, xal ix- 

xeie9ca rtg tv&ita r} K, xai ytyovita wg (iiv ij BF 

wpos ^iiv rH, ovtag ij K repoj r^w j4, ag Ss rj idF 

20 Kpog T^v TE, ovraq ii A spog zr(ii M. 

Ot aga Xoyoi t^s te K wpog njv A xal t^g 
Kpos Tij»' M ol avroi slet tois Xoyots tav tcXevqi 
zije %a BF «pos tijv FH xal tijs ^T tcqos ziiv FK 
tiXX' 6 r^s K jTpos -'^ Xoyos evyxeitai i'x zt roij 
26 T^s ^ ^9^S ^ X6yov xai roi r^s -^ «pog M' mazs 
xal i} K TtQos T^i' M X6yov Ix^i tov evyxeifisvov 

SSin. Tteon in Ptoiem, p. 386. Eutoc. in ApoUon. p. 38, 
id. in Arshimed. 111 p, 236, 23. 



J 



l.ueiimv — it itiitmvSQa}inKtt.bitaj6.F. 1. PJ7]np: 

pn] r- " > TT- -rf __■_ __•..■. _„•- T,r._ 



2. pn] np p. T^ ns v. xpos t^*] «pds ] 



J 



ELEMENTORUM LIBER VI. 147 

XXIII. 

Parallelogramma aequiangula inter se rationem ex 
rationibns laterum compositam habent. 

- Sint parallelogramma aequi- 



f 



■i 




angula AF, FZ habentia 
L BFJ = ErH. 



\T p dico, parallelogramma AFy FZ 

rationem ex rationibus^) late- 

/ I rum compositam habere. 

^ ^ ponantur enim ita, ut in 

eadem recta sint JJT, FH. itaque etiam ^F, FE in 
eadem recta sunt. et expleatur parallelogrammum 
dHy et ponatur recta K^ et sit 

Br:rH = K:A et Ar:rE=A:M. 

itaque rationes K : A et A : M eaedem sunt ac 

rationes laterum, Br^TH et Ar^TE. sed K:M 

= K: Ax A : M. quare K Rd M rationem ex ratio- 

nibus laterum compositam habet. et quoniam est 

1) '£x tmv nXsvqmv per totam propositionem neglegentins 
dicitnr pro in t6v z6v nXsvQmv {Xoyoov); sed cum semper ita 
in codicibus traditum sit et idem apud Theonem et Eutocinm 
sematum sit, de errore librarii cogitandum non est. 

P7I] nP P. T^v] om. BFp. ovTosl om. BFjp. 4. ti]v] 
om. BFp. JTP] P, V m. 1; PJT Bp, V m. 2, F? /ic^fov 
a^a] bis p. 6. fAfi-Jov F. 6. GN] N e corr. m. 2 V, 

eras. F. 7. H@] @H P. &Qa iativ P. 8. hs' p et deleto 
g F. 11. taov V, corr. m. 2. 12. ETH] mutat. in EFG B. 

13. rZ] in ras. m. 1 V. 14. nXsvqmv] P; nXsvQmv tov 

ts ov ^x^i j5 jB r (corr. ex FJB p) nqos FH (t^ FH V, FH 
mutat. in rO B) nal tov ov k'xsi ri jrnQOs FE (triv FE V) 
Theon (BFVp). 16. FH] mutat. in r@ B. iativ B. 17. 
JH] mutat. in J0 B. 18. K] post ras. 1 litt. F. 19. 

rH] mutat. in r@ B. 21. t>}vJ om. BFp. 22. ti^v] om. 
BF^. statv PF. 23. trjv] om. Bp. r/f] mutat. in FG B. 

T?}v] om. Bp. 

10* 



148 STOrXElilN 5'- 

ix rav xlivQtBv. xkI insi ienv tog i} BV apog t^v 
r//, owcjg ro AF nccQaKhiloypttfiiiov npbg zo r&, 
akk' fog ij Br jrpog xi\v rH, owojg »; K Jrpo? Tiji' 
y^, xaX ms apK ^ K «poj t^v ^, oSttJs ro AT jrpog 
5 to r"®. TiKhv, imi isziv mg ^ ^^P rcpog t^h r£, 
ovTCjg to F@ i^rapa^Ajj^o^^pKfifioi' srpog lo fZ, aAA' 
(3? ^ ^J^ 3rpog r^v Ffi, ovro^ ij A «pos rijv itf, 
xai (05 Kpa ^ ^ n^pog ttjV M, ourtBg xb F® xa$<x.i.h]K6- 
yQa(i[iov ngog ro FZ naQalXrjl6yffttii[iov. inel ovv 

10 iSsCx^Vf '^^ f*^" ^ ■'^ "^pog rijv A, omiag ro ^f WKpa^- 
AijAoypttfi^ov JTpog ro J*® :ro!paA^i}Ao'^pffi^^ot', rog 3s ^ 
ji rfpog ri\v M, owreg ro F® TtKQttXktiXoyQafifiov jrpog 
To rZ na^ttXXijXoypttfiftov, di teov aQa ierlv atg ij 
K JTpog r^f M, ovrrag ro AV itQog ro FZ jrapa^AijAo- 

15 ypwftfioi'. ij tf^ Jf srpDg t»jv Af Aoyoi' fx^( 
y,£vov ix rmv xXtvpiSv' xal to AF apa ;rpog 
Aoyov ^x^i Toi' avyxeiiitvov hi rmv nXtvQav. 

Ttt &pu iaoydviK aKQttXXi]X6yQa^(itt ngoq aXki!\ia 
koyov %M Toi' evyxaifisvov ix rtov nXEveaV onsg 

20 ^Sii dEt^tti. 

xd'. 
Uavrog jiaQalkiiXoyQtt(i(iov ra ffept rijv dt\ 
(iBTQOv ^cegttXXrjX6yQtt(ifitt o^oia i6zi Tfci 
oAn xttl aAAijAotg. 

25 "Eotra jtapaAiijAoypaftftow to ABF^, dtafietffos' 
avTov ij AF, itsQl 61 rijv AT JtapaAAijAdypaftfta laita 
ra EH, ®K' A^yo, ori £xa'r£pov rtoc EH, &K nttffakXrj- 
Xoy(fdfi.(iaiv o(ioi6v ioxt. oXa ra ABFJ xai «AA^Aotg. 






I 



ritj 



ELEMENTORUM LIBER VI. 149 

Br\rH=Ar:r® [prop. I], et Br.rH — Kiji, 
drit etiam K : A ^ AF : F®. rursus quoniam est 

jr :rE^r@:rz [prop. I], et ^r : FjE; = -^ : iW, 

erit etiam A: M= FO : FZ. iam quoniam demon- 
Btratum est, esse K : A = Ar:r@ et A : M= rs 
: rZ, ex aequo [V, 22] erit X : Af = ^F: TZ. sod 
K aA M rationem ez rationibus laterum compositam 
habet. quare etiam AF ad FZ rationem ex rationi- 
bus laterum compositam habet. 

Ergo parallelogramma aequiangula inter se ratio- 
nem ez rationibus laterum compositam habent; quod 
erat demonstrandum. 

XXIV. 

In quouis parallelogrammo parallelogramma cir- 
cum diametrum posita similia sunt et toti et inter se. 

Sit parallelogrammum ABFAy diametrus autem 
eius AFy et parallelogramma circum AF posita sint 
EHy &K, dico, parallelogramma EH, ®K similia 
esse et toti ABFA et inter se. 



mutat. in TS B. 4. to] 17 p. AT'] AK e corr. V; T 

mntat. in i^ m. recentissima p. 5. rd] xriv p. T®'\ mutat. 
in Ftf B; T mutat. in z/ m. recentiss. p. . 6. FOJ mntat. 
in TH B. 7. t?5v] om. BFp. t^v] om. P. 10. 17 

ykiv p. 11. T&l mutat. in TH B. 17] to qp (non F). 

12. re>] mntat. in E0 F, in TH B. 14. AT] PV; AT 
naQ€ilXTjX6yQafi,fiov Bp et comp. F. In fi^ra litterae H. S 
in B permutatae snnt a m. l^ sed mutationes in textu nuc 
BX>ectimtes a m. 2 uidentur esse. 16. &(fcc] m. >2 V. 17. 
cvY%Bifiiv(ov P, corr. m. 1. 21. xf Fp. 28. iattv PB; 

comp. p. ^27. EH] (alt.) in ras. F. 28. iativ PBF; 

comp. p. oXcp] m. 2 V. 



150 



STOIKEIiiN ! 



\ 



'EsbI y&Q TQiytovov tov ABF Jtapa (ilav rav 
TtXBVQNv zijv BV TJKfKt 1] £Z, avaXoyov iottv «s 
1) BE Ttgog rijv EA^ ovraq r} FZ jrpog tijv ZA. 
itihv, BTtel tptj^mvou roG j^f^ rcapo: fii^av t^j' Zii 
6 ijxtftt ij Zif, avuXoyov iativ rog ^ FZ n;pog T^i; 
Z^, ovrrag ?/ ^H ;tpos rijv W^. aAA' ag ij FZ 
JtQog T^v ZA, ovTcog iSsC%&7i xal ii BE irpog t^v 
EA' xrI ms «pa ^ B£ ;rpos ti\v EA, oilrojg ^ 
^if Jtpog rTjv i/j4, xai aw&ivri ntpa ojs »/ Bv^ ^pog 

10 AE, ovtas ij ^.^ JiQog AH, xa\ ivakKrt^ la; ^ BA 
jrpoj T^v y*.^, ovrwL; ^ Ev^ ;rp6s tijv AH. tav 
aQa ABF^, EH itKfiaXKiiloy^afiiiav avaloyov tiaiv 
al TtkBvQal ai refpi r^v xoivijv ymvCav tijv vith BA/t. 
xai inBi nccQtiXlrjlog iotiv rj HZ tfj z/f, fffTj iatlv 

15 jj (thv vitb AZH yavCa rjj {lao ^FA- xal xolv^ 
tiBv Svo t^iymvav rdv AdF, AHZ ri vno AAV 
yavia' itSoycoviov «pa iGrl to A^F rptyojrov Tp 
AHZ tQiytova. dia ta avta dij xal to AVB tQCya- 
vov laoyavtov idti ra AZE tpiymva, xai oXov ro 

20 ABFJ napaAAijXdypa^^ov tp EH aagaXlyjXoyQanfioi 
iaoymviov iativ. avdXoyov «pa iatlv dtg tj A^ n^o^ 
tijv jdr, ourojg ti AH JtQog rijv HZ, ag di ij ^F 
TtQog ti\v FA, ofJtcjg ij HZ wpog r^v ZA, ag Sh t] 
AF TtQog tijv FB, ovtag i] AZ jrpos t^i' ZE, xal 

25 iri (wg ij FB itpog tIjv BA, ovtiag rj ZE ngog r^v EA. 

2. T^v] in raa. m. 2 V, corr, ex z^ m. 2 P. EZ] HZ 
ai. rec. p. 3. BE] nmtat. in BH m. rec. p. EA] mntat. 
ia H^ m. rec. p; BJ tp. 1. JFJ] PF, V m. 1; ,)jr 

Bp, V m. 2, 5, Zff] mutat. io ZE m. reo. p. 6, J H] 

mutat. in JE m, rec. p. 8. E^] (prins) EJ q> (dou F). Seq, 
in p: ovrcos ^ ilH jrpos ")v HA Kal flwii&iiTi 00«, del, m. 1. 
ovtag tal p. 9, aaa] om. P. 10. t^v ,4E V. ourcoe] 

om, BFp. t^* AH V. Bij] vlB p. 12. osb] P; om. 




ELEMENTORUM LIBER VI. 151 

nam quoniam in triangulo ABF uni lateri BF 
parallela ducta est EZ, erit BE : EA =^ FZ : ZA 

[prop. II]. rursus quoniam in tri- 
angulo AFJ uni lateri FJ parallela 
\fi ducta est ZH^ erit 

rZ : ZA = JH:HA 
^ [id.]. sed demonstratum est, esse 
rZ : ZA = BE : EA. quare etiam 
BE: EA = JH: HA, et componendo [V, 18] 

BA:AE=JA:AH, 
etpermutando[V,16] BA:AJ = EA:AH, itaquelatera 
communem angulum BAd comprehendentia parallelo- 
grammorum ABF^y EH proportionalia sunt. et quo- 
niam HZ rectae z/F parallela est, erit LAZH= ^FA 
[I, 29]; et duorum triangulorum AJr, AHZ com- 
munis est L ^AF, itaque triangulus AJF aequi- 
angulus est triangulo AHZ [I, 32]. eadem de causa 
etiam triangulus AFB triangulo AZE aequiangulus 
est; et totum parallelogrammum ABFd parallelo- 
grammo EH aequiangulum est. itaque^) erit 
AJ:Jr=AH: HZ, ^t: FA = HZ: ZA et 
Ar : TB = AZ : ZE, rB:BA = ZE:EA [prop. IV]. 

* 1) Hoc apa lin. 21 non ad ultima uerba, sed ad prozime 
antecedentia lin. 17 — 19 refertur. 

BPVp. EH] E postea insert. F; deinde a^a add. m. 2 BPV. 

13. «n (alt.) om. F. 14. i^ari'] tari Ss F. 15. AZH]F; 
AHZ Theon (BFVp). yovta] m. 2 V. t^]_P; tv vno 
AJr rj Sl vno HZA {ZHA F) t^ Theon (BFVp). 16. 

AHZ] PF, V m. 1; AZH Bp, V m.' 2. 17. yov^a] om. Bp. 

t6 AJr] P, V m. 1; om. F; t6 JAT Bp, V m. 2. 18. 
AHZ] litt. HZ e corr. p. AFB) ABF V. 19. olov] olov 
Hoa V. 20. lcoymviov icti tm EH fCccqoclXTjXoYffcififMp V. 

26. EA] AE, eraso E F. 



.. i.ua***«rfr>. - 



152 



STOIXEIP-N ?■. 



1 



Ttas ^ HZ jrpog t^v Z^, tog tfe ij AF a^og zijv 
FB, ovxag ij AZ Jipog ^151/ ZE, dt' l6ov kqu iszlv 
aq rj ^T n(fbg zijv FB, oiircjs r] HZ XQog xr{v ZE, 
6 T(DU apa ABF^, EH nagtt?,?.i}ioyQuiiiiav avdloyov 
Eleiv ttt nXEVQoi aC ni(/l rag i'aag ymviag' ofioiov aiftt 
ietl t6 ABV^ jrwpttAijjAdypa/ifiov tw EH xaQaii.ijlO' 
ygdliii^. 6itt t« ainu Sij t6 ABFJ napaA^^Ao- 
YQayifiov xal tcj K& iiaQalii]ioyQK(ifia oiiotov iativ 

10 ^xatffioi' n(ia tcd)' £H, @K izaQaXXTjkoyQaftfiav tm 
ABF^ [wKpaAAijAoypKfijtj)] o^otoi' ^flTtv. t« di tp 
avra iv^vyQafifia o^otc xui a^A^Aotg ioTiv oftouL'^ 
xai to £/f RQtt JiaQal^r)X6y(fttfi[tov rm @iir nnpai 
Xi^}.oyQaftnei oftotoi' ioziv. 

l& Tlttvzbg apa jtaQai.3.tiloyQdfi(tov za ^iqI t^v dta 

fxiTpof irapaAAij/o^^^rfftfta o^ota ^ari tra te 0^9 3 
aWijilois' oacQ B^ti det^at. 



Ta So&ivzi £v9vyQdiiiia ofioiov xal 1 
3 ta do&evti teov tb avtb avdfqeaG^ai. 

"E0za tl ftji/ So&iv sv&vyQafinov, m Stt oftou] 
ovazr]<saa&ai, to ABF, a di Sii i6av, 16 ^' Sst i 
tw (ihv ASr ofiotoi', za ii ^ Isov t6 «uto 0tiffT]9 
ma^ai. 



XXV. Hero def. 116. Eotooiafl in ApoUon. p, 63. 

1. r^] r eras. F. 2. HZ] ZH Fp. AVA erae. ] 
3. rBl B eras. F. i. rBl sr P. 6. E^o.»'] ef- eraB. I 

7. 10] corr. ei tra m. 2 V. wceailijjloyeBftfiov] corr. 
nafiaJUtlllaypa^^ca m. 2 Y. t^] corc. ez to m. 2 V. Rcfpnl- 
Ijjioypafifio» V, 'corr. m. 3. ' 8. Sr\\ gii xtti F; xai add. 
T m. 2. 9. «H^] m, 2 F. K»} SK P. 11. ««(«Ui)- 



r 



ELEMENTORUM LIBER VI. 153 

et quoniam demonstratum est, esse zJF: FA = HZ 
: ZA et AV: FB -^JZ: ZE, es aequo -erit [V, 22] 
^r : PB = HZ : ZE. ergo in parallelograimnis 
ABF^, EH iatera aequaies anguloa compreheudentia 
proportionalia Bant,') itaque ABF^ '^ EH [def. 1].') 
eadem de causa etiam ABF^ ~ K&. itaque utrumque 
parallelogrammum EJd, &K parallelogrammo ABT^ 
simile eat. quae autem eidem figurae rectiliueae 
aimiles sant figurae, etiam inter ae aimiles sunt 
[prop. XXI]. qiiare etiam EH"-&K. 

Ergo iu quouis parallelogrammo paraUelogramma 
circum diametrum posita similia sunt et toti et inter 
Be; quod erat demoQstraudum. 

XXV. 

Datae figurae rectilineae similem et alii figurae 
datae aequalem eandem figuram conatruere. 

Sit data figura rectilinea, cui similem figuram 
oporteat construere, ABT, cui autem aequaiem opor- 
teat, ^, oportet igitur figuram construere figurae 
ASr similem, figurae autem z/ eaudem aequalem. 

1) Nu.m demoDBtrauimus BA : AJ ^ EA:AH {$. 160, 10), 
Ail : j:ir -' AH : HZ (p. 160, 21), HZ : ZE = JT : FB 
(lin. 4), ZE ! E.i = TB : BA (p. 150, 26). 

2) Nun etiam aequiausula saat [p. 160, SO). — hac ratione 
diluuntor, opiuor, canillatiorieB Simaoni p. 378; qnamqnain 
confitendum est, Euclidem hic nomiihil a solito ordine dilucido 

loyeatifiio] om. P. ^oriv] F, comp. p; iaii PBV. 12. 

iniy'] liaiv V. 13. Sea] om. p. Bk]» in raa. V. 14. 
lou*] comp. Vp! foiiPBF. 16. tflm. 2F. 18- k?i' Fp. 20. 
avvetijeaaifai P; coir. m. rec. 21. Post m eras. Se B. 22. ovii- 
aiTeaa»at P; corr. m.rec. Si dtf ftfwjmras. m. 2 V. 28. toj] 
(prins) corr. ex td m. 1 p; iti F. evvaf^<ieia9<ti P; corr. m. rec. 



154 



ETOrXEiaN 5 - 



raw ■ 



TTaQtt^t^Xrju^ci ya^ nitQa ^lv xi}v BF rra j^BF 
TQiycovfp (aov mcQaAlr)l6yQa(i{iov to BE, naQ& di 
riiv FB Tip d i'eov itaQaXXtjXoyQaftfiov ro FM iv 
ymvia rij vno ZVE, tJ ioriv l'9ri r»j vno FBA. i% 

5 tv^eiai aqa iarlv ^ filv BF rij FZ, i; 6i AE nj 
EM. xa\ elkri<p9a rmv BF, FZ fiderj avaXoyov ^ 
H@, xal avaytyffttip&o} ano T^g H& rp -ABr Sfioi 
TE xal Ofioimg xsifiEvov ro KH®. 

Kal ixiC tariv cSs ^ BV irpoj ri^v H&, ovras 
10 7} H® itpos tijv rZ, ittv 8% rpEfs Bv&ttai dvaXoyov 
aeiv, ieriv atq ij Jtpolrj? ffpog Tijv rQirtjv, owroij rl 
ttTto T^s «pojTijs ilSog itgog to «jeo Tijg dsvtSQas to 
ojtoiov xal ofioias dvayQafpofitvov, letiv apa ebg ^ 
Br JTpos rijv rZ, ovttog ro ABV rQiyavov JtQos 

3 To KH& TQiymvov. dXla xal lag ^ BF XQog rijv 
rZ, oSrojg ro BS ^rapaAAjj^oypafifiov Kpog t6 EZ 
jTftpRAATjAoj^pa^^ov. xai rog «po; to ABF rgiyoivov 
jtpog ro KH@ tQiymvov, oiJrias to BB }T(;f{)aAAi]<ld' 
yQttfiiiov Kpos To EZ ffc:(io:'AA}jAo}'pafifiO]'' ii/«>ljln:| 

Q upa Gig t6 ABF XQiycovov irpos rf 5£ ;rapo!iA»;Ao- 
yQaniiov, ovtos ro KH& TQiyavov xqos rb KZ 
aaQtti.i.ri3.6yQa(i(iov. Heov 6% rb ABF TQiympov ra 
BE naQalXri?.oyQtt[iiim' fffov apa xal ro KH@ tqC- 
ymvov To EZ TtaQuXXiiXoyQdftfim, dXXd ro EZ itaQ- 
26 aXXijXoyQafifiov ra A ieriv teov xal to KH@ Sga 
ietiv leov. i^ffrt (Je to KH@ xaX 



r^ 

O^OIOV. 



1. Tw .IBr] sapra F. i. FB^] FB^ qs. 5, BF] 

Pgi, Vm, 1; TB Bp, V m. 2. 6. xal Ftttjqjfl-a)] «ewd^- 

q*i» qi post rw. 7. H9] (priua) eraa, F. m] td F. 



(pa So&ivri tv^vyQttfifi^ rm ABT ofii 



1 



ELEMBNTORUM LIBER VL 165 

Nam rectae BF triangulo ABF aequale adplicetar 
parallelogrammum BE [I, 44], rectae autem FE 

jL ir 




figurae /1 aequale parallelogrammum FM in angulo 
ZFE aequali angulo FBA [I, 45]. itaque BF, FZ * 
in eadem recta sunt et item AE, EM. et sumatur 
rectarum BFy FZ media proportionalis HS [prop. XIII]; 
et in H0 triangulo ABF similis et similiter positus 
construatur KHS [prop. XYIII]. et quoniam est 
BF: H6 = HB : FZ, et si tres rectae proportionales 
sunt, est ut prima ad tertiam, ita figura in prima 
descripta ad figuram in secunda similem et similiter 
descriptam [prop. XIX coroU.], erit 

Br:rZ = ABr:KH@. 
uerum etiam BF : FZ = BE: EZ [prop. I]. quare 
etiam ABF: KH& =^ BE:EZ. permutando igitur 
[V, 16] ABr:BE= KHG:EZ. sed ABF—BE. 
itaque etiam KH@ = EZ. sed ^Z = z/. quare 
etiam KH0 = ^. erat autem etiam KHS <^ ABF. 
Ergo datae figurae rectilineae ABF similis et 

8. T€] om. V. 10. ri] eras. F. 11. ^ffrt»] om. P. 16. 
TQiymvov] om. V. Supra BF scr. ^disig et supra FZ lin. 16 

edatv m. rec. P. 17. xal mg &Qa — 19: nttQaUriXoyQannov] 
is p; corr. m. 1. 19. EZ] ZE p (sed in repetitione EZ). 
25. taov %cci] in mg. transit F. KH9] in ras. m. 2 F. 

uQa tm J icTiv taov] om. F. 26. Im dl ro] q> cum ras. 

2 litt. ante rd. 




156 STOKEiaN ff'. 

xal aiXa ta So&ivrt ip ^ Itfov i 
To KH@' ontfi tSii aoiijetti. 



'Ettv «Ko itaQaX^i}loyp«n[iov sta<fai.}.i]Xi 
& ygafifiov uipaiife&ii onoiov re r^ ol<p Kal 6(io£t 
xe£(i£vov xotvijv ymvtav ixov «vip, Wfpl 
avxriv SittfieTQov iati rc5 oAp. 

"^jro yap lECfpnfAATjAojrpfCfi^ou tou ABFjd %aQtt} 
XTjXoygafiiiov aipriQ^a9a ro ^Z ofiotov ra ABF/l 
10 xa^ bfioitaq xt(fisvov xotviiv ytoviav bxov avrip ripi 
vno zlAB' liyco, Sri aeffl rijv avrijv Sittfiez^v 
t6 ASr^ T(5 AZ. 

Mij ya^, aW si Svvaxov, emto [avvav^ 
fitTQOS rj A&r, xal ix^iTj&etea rj HZ Si^x^^ 
15 To ®, xal ij^tffci Siic roiJ © OTioTiqa tmv A.J, BV 
xafittXXijXog ^ @K. 

'E%el oifv 3t£pl r^v avtijv SiafitTgov iett ro ABFJ 

tp KH, ienv apw rog ^ ,JA wpog tiiv AB, oBroje 

)) HA Jtpog r^v jtfif. Jffrt Si xal Sia r^v oftotoTtjTa 

20 Tcov ABFjd, EH xttl ag ij AA jrpog r^r AB.^ oiicus 

ij ff^itf Jtpotf xr(u AE' xal tog Kpa ^ H^i Jtpo? r^w 



ripi 

9 



m. Theon ^EFp, V m. 1). cwi- 

,. _. ,._ _ ,. _. «cpalUj;loy(i(ifio» P. B. aqjcn- 

(fd^v qd. i^ oilai] 10 oHov qn iil ta.», 8. naifuXl,j]Xoyi}iiif 
fiovyatP. 9. /Z] supra 2 litt. eras. auat in V; JEZHBp. 
Ti»] iD ip. 11, iextv F. 12. id] za V, corr. m. 2, 

ABFJ V. 13. ««ri»] om. FV. 14. A&F} tp; oa inter 
dnab tas. F. ««1 {n^Xrj^Elaa — 16; to 0] P; om. TLeon 

(BFVp). 18. Poat KH add. Theoc: Ztioiov lau x6 ABT^ 
iffl KH (BFVp). 21. «k1 106 ^ga — p. 168 1: nrpoe i' 

.i£] om. Bp. HA] "A-HJ. 




ELEMENTOBUM LIBER VI. 157 

alii figurae datae ^d aequalis eadem donstructa est 
figura KH@] quod oportebat fieri. 

XXVI. 

Si a parallelogramino aufertur parallelogrammum 
toti simile et similiter positum et communem angu- 
lum habens, circum eandem diametrum positum est 
ac totum. 

Nam a parallelogrammo ABF/I auferatur paralle- 

logrammum AZ simile parallelogrammo ABFA et 

similiter positum et communem habens angulum jdAB. 

dicO; ABFA et AZ circum eandem diametrum po- 

sita esse. 

ne sint enim; sed^ si 

fieri potest, diametrus sit 

ASr}) et producta HZ 

ad ® educatur*), et per 

® utrique AA, BTparal- 

lela ducatur @K [I, 31 

et 30]. iam quoniam 

ABFd et KH circum eandem diametrum sunt po- 

sita, erit AA : AB = HA : AK,^) sed propter 

aimilitudinem parallelogrammorum ABFAj EH etii 

etiam [def. 1] AA : AB = HA : AE. itaque etiam 

1) Debuit ita dicere: nam ai AZF diametras parallelo- 
grammi AF non eet, sit ASr. adparet, avtcoy lin. 13 ferri 
non posse, sed malim com FV delere quam cum Feyrardo in 
avtov corrigere; glossema sponte et in P et in Theoninis 
nonnnllis ortnm esse potest. 

2) XJerba xorl inplri&SLCa cet. lin. 14 — 15 om. Theon, quia 
in fignra codd. permutatae sunt litterae E, Z et K, S; cfr, 
p. 168, 3. ego cmn Augusto his uerbis retentis errorem 
p. 168, 3 et figuram corrigere malui. Campani figura nostrae 
similior esi 

3) Nam similia sunt (prop. 24); tum u. def. 1. 




158 



ETOIXEIHN S'. 




-■^K, ovzag ij HA jrpog rijv AE. r\ HA apa i 
ixarigav ztav AK, AE t6v amov exh loyov. j^oyj 
liga ietlv ii AE r^ AK ij iXttrrav rfi fiii^avf oxeQ 
icrlv aivvarov. ovx aQtt ovx ieri ji£qI rijv avrijv 
b dtufierffov t6 ABFi^ %a AZ' :ic^\ rijv nwr^v opo 
ietl Stafisrgov to ABT^ jrapaWijioypKfi^ioi' ru AZ 
TtapaXX^^Xoyptiitfia. 

'Eav uQu ano xaQaXlr}Xoy^a^fiov nagaXlt^loypttfi- 
ftof tt<paips&^ oiioiov Ts rto oX^ xal ofioicos xeiftcvai 
10 xoiv^v yavCav i^ov avra, nsgl r^v avzijv StaftErM 
isrt zm oAk)' o:t€Q idst Set^ai, 



HdvrtQV rav TcagK rijv avrijv sv&Btav siapt^ 
^aXXofiiviov nagaXlTjXoyQatifiaiv xal iXXst} 

15 rtov iHdeat naQaXXTjXoypafifiots 6[ioiois t£ xal 
ofioias XEtfiivoig ra dxo tijs ^iiteeiag 
yQttfpofiiva ftiyiaxov iort to dnt tijQ ^fitffeA 
aaQtt^aKXofisvov [icagaXXtjXoyQafinov] ofioH 
ov ta iXltlfifiKTt. 

20 "Eera Ev&eta i] AB xal Tir(i^aQa SC%a xara th 
r, Xttl jtttQtt^E^X^e&co Jtttiftt i^v AB tv&flav t6 AJ 
TtapaXXijXoyQaiiiiov iXXtinov sHdtt TiapaXXtjXoyQdfiiie) 
ra JB uvayQttiphVTt dnb t^g ^fiteiias tijs AB, rowrsUTt 
Trjs VB' Xiya, ort ndvrttiv rav JtttQK tijv AB nafftt- 

-5 ^aXXofiivav JtagaXXTjXoyQdfiiiav xal iXXctnovrav ttStgL,- 
[nttffaXXTiXoypdftfiots] o^o^tg ic xal oftoias xsifievtili^ 



1 



1. AK] Pi AEK, E in ra 

corr. m, rec; AK V. Squ] i 

Vm. 2. AK] AB PBFp, ^i 

4. ov*] (alt.) om. BVp. loi 



AE V. 



AE} AB *7 
^Ej AK PFBp, 
Dv P, oorr, m. S. 
5. AZ] P91; Ae 



ELEMENTORUM LDBER VI. 159 

HA : AK = HA : AE, ergo HA ad utramque 
AKy AE eandem rationem habet. quare AE — AK 
[V; 9] minor maiori; quod fieri non potest. quare 
fieri non potest^ ut ABF^, AZ circum eandem dia- 
metrum posita non sint. ergo parallelogramma 
ABFd^ AZ circum eandem diametrum posita sunt. 
Ergo si a parallelogrammo aufertur parallelo- 
grammum toti simile et similiter positum et commu- 
nem angulum habens; circum eandem diametrum po-. 
situm est ac totum; quod erat demonstrandum, 

XXVII. 

Omnium parallelogrammorum eidem rectae adpli- 
catorum et deficientium figuris parallelogrammis simi- 
libus et similiter positis ei^ quae in dimidia describi- 
tur; maximum est parallelogrammum dimidiae ad- 
plicatum defectui simile. 

Sit recta AB et in duas partes aequales secetur 
in F; et rectae AB adplicetur parallelogrammum AJ 
deficiens figura parallelogramma z/£ in dimidia rec- 
tae ABy boc est in rB, descripta. dicO; omnium 
parallelogrammorum rectae AB adplicatorum et figuris 

BVp. 6. kcxCv P. 10. ^%ov ycavCccvY, avTijv] supra m. 1 p. 

12. X' Fp. 17. TS iatt p. 18. naQaXafA§av6fiBvov P; 

corr. m. rec. naoaXXriloYQafkfiov] m. rec. F. oft,oiov] corr. 
ex oft,oi P. 19. ov t<p] ov to 9 in ras. iXXstfiati p. 21. 
Ti}»l triv avtriv P. AJ] d in ras. m. 2 V; AB 9. 23. 

^BJ JS tp (non F). Post hoc uocab. add. Theon: ofjMlm xb 
xttl OftoC(oq avayqatpivti (F; pro ofkoCm Bpqp, V m. 2 nab. 
ofi^oiov'^ pro avayqafpivti, Bp: avayQa(piv, v tisifisv seq. ras.; 
-T* in F punctis del.). dvayQatpivtt] P; t^ Theon (BFVp). 

rjfiiasiag] rjfiiasCag dvayqatpivti FV. AB] AJ q> (non F). 
rovxiativ P. 26. «rdew] 9 (aliud uerbum habuit F); sCSsaw P. 

26. naQaXXrjXoy qdfifiOLgy om. P. 



160 ETOIXEIGN S'. 



E/3A^««^yS^ 



xapa t^v A B iv&etav to AZ Tiapallrii.oypaitHM' 
ilXtiTtov st6ci iiaQaXXTjloyQtt^ne} rra ZB ofioia u 
kkI 6fioiB3s xBifiBvci rco ^B' liya, ori (tft^ov iori to 
h Aid rov AZ. 

'E^el yag ofioiov iaxi to idB JiaQakltiloy^afifiov 
ra ZB 3ra(fall7ji.oyQKjific), jrepi njv kvt^v etfft iJta- 
fittpov. ijX'^ro avrSv SielfiSTQog i} jJB, ical xartt- 
yEyQafpQ^m ro OxVf''^- 

10 'ExbI ovv Caov iaxl to VZ rp ZE, xoivov di 
to ZB, okov aQu zh FB oXa rto KE iativ faov, 
ttXXa ro r& ta FH iativ leov, iael xal ij AF tr^ 
FB. Xttl ro Hr «pK ta EK leriv iaov. xoivov 
nQoaxtie&co ro FZ' oAov uqk to AZ rp AMN 

15 yvwftovl ietiv fffov SetB r6 JB TtKQaXlr^XoyQait- 
ftov, tovrieri, t6 AA, xov AZ naQalltiXoyQKfiftov fitt- 
lov ietiv. 

navtmv uQK r(ai' xaQa rijv avriiv BV%Etav JiaQK- 
^KXXofiivmv naQaXf.7]XoyQa(tfimv xal iXkBixovtmv el'Seei 

20 naQaXXriJ,oy(faft^oiq oftoioig re xal 6ftolcas xttftivoig 
rw nreo rijq ijfiiaeias avayQatpoftiva fisyiavov iati x6 
eati riig ijfiieBias TiaQa^Xij^iv a%EQ iSei dEt^ai. 



1. i^] To F. vaeapel-^<t9ia p. 2. AB]^ B e cow, m. 
1 p. 3. aagallTjloyeditm p. 7. ncpl apv ttJv Bp. 10. 
foo»] eupra m. 1 V. ZE] corr. ex Z» m. rec. P. ff^] P; 
jtpoffK£i'<i9ra Theon (BFVp). U. TQ] e corr, P m. reo, 

KE] corr. ei K9 m. rec. P. 12. r&] oorr. ei FE P m. reo. 

13. rs] PF; htiv ieri anpra add. V; TB fini ^iwi'»' Bp. 
EXJ e corr. P m, rec. 14. olo*] aeq, ms, 3—3 litt. F. 16. 
j<Z] inter ,,< et Z raa. 1 litt. F, 17. itfri B. 18, avt^v] 
om. p. 19, naQctl.lTiKiyiiia^^tDV — 82; dfr|o:i] nal ra f^^c p. 

22. Sei%Bi] seq. in omniljQB codd. demonatratio alia, qnam 
in appendicem reiecimusj u. p. 161 uot. S. 




ELEMENTORUM LIBER VL 161 

similibus et similiter positis figu- 
rae z/£deficientiummazimum esse 
Ad, adplicetur enim rectae AE 
parallelogrammum AZ deficiens 
figura parallelogramma ZB simili 
et similiter posita figurae z/5. dico, esse AJ^AZ. 
nam quoniam dB f^ ZB, circum eandem diame- 
trum sunt posita [prop. XXVI]. ducatur eorum dia- 
metrus ^By et describatur figura.^) iam quoniam 
rZ = ZE [I, 43] et commune est ZB, erit F® = KE. 
sed r® = FH^ quoniam AF = FB [prop. I]. quare 
etiam HF = EK, commune adiiciatur FZ. itaque 
AZ = AMN. quare ^B > AZ, h. e. AJ > AZ, 
Ergo omnium parallelogrammorum eidem rectae 
adplicatorum et deficientium figuris parallelogrammis 
similibus et similiter positis ei, quae in dimidia 
describitur, maximum est, quod dimidiae adplicatur; 
quod erat demonstrandum.^) 



1) H. e. producantur HZ ad et KZ usque ad dE\ 
cfr. II, 7, 8. 

2) Itaque is solus casus tractatur, ubi AK^ AF, nec 
opus est alterum, ubi AK < AF^ propria demonstratione 
ostendere nec hoc moris est Euclidis. sane in codd. omnibus 
additur demonstratio huius quoque casus. sed apertissime 
interpolata est; nam primum ante lin. 18 sq., non post eas 
inserenda erat, deinde iam ab initio in praeparatione duo 
casus respiciendi erant nec hoc unquam neglexit Euclides, ubi 
plures casus habet; ita etiam in altero casu eaedem litterae, 
quae in priore, usurpatae essent, quod iure postulat Simsonus 
p. 380. Campanus VI, 26 utrumque casum demonstrat. 



Euclides, edd. Heiberg et Menge. II. 11 



ETOEKEIQN S 



Xt) . 




IlaQa rijv So^sMav tv&siav rra doQ-evti^ 
sv&vyQafifim (Bov xaQaAXrjAoyQamiov itaQa 
^akatv iX^sinov etdti nagakXijloyQafijip ofioip 

> rp do&svTf Set Ss ro Sidofi-svov ev&vyfafifiov 
[cj Ssi i'aov 7taifttPaXeiv'\ jt^ iieii,ov elvai loi; 
«jEo rijs yj(iiaaias ttvayQacpo(iBvov onoiov ra 
iXleiftfiatL [tov ts uTto z^g il[iieEiag xal ra Sei 
ofiOtov ilXeiiteiv]. 

"EdTa ii (iiv do&eiea ev&eta ij AB, xo Se So^ev 
tv9vyQa(i[iov , a Sti leov naQa tijv AB aaQafia^eHVy 
ro r [i^ fiei^ov [ov] rov anb r^g ijfiiSeCug r^g jiB 
avaygaqiofiivov ofioiov rm ii.i.eCfi(tarL, <p 5\ Set ofMtot 




ili.eCxEiv, To /i' Set Si) tckqk zijv So&eteav ev&tte 
I 16 ^ijv AB Tfi So&ivti ev^vyQa^yiOi rra F ieov naQeck- 
ItiXoyQaftiiov iiaQa^aXstv ekkelTiov etSsi nagailiiliiy- 
yQa[ifta oftoCc) ovri Tto j^. 

Tetff^e&m i} AB SC%tt KKta ro E ej](ietoVf 
avayeyQag}&a aito tijg EB ta ^d o(ioiov xal ofioi 



1, xij'] om. F; X^' p. 2. £v9i[av'} mg. 

naeaeaUeiv V. 5. ip] Ofri im V. Si} J^' PBFV] 



FT^l 



m ELEMENTORUM LIBEB VI. 163 

xsvni. 

Datae rectae datae figurae rectilineae aequale 
parallelogrammuin adplicare deficiens figura parallelo- 
gramma datae aimili. oportet autem, figuram rec- 
tilineam datam') maiorem non esae figura in dimidia 
recta descripta defectui aimili.^) 

Sit data recta ^B, et data figura rectilinea, cai 
aequalem figuram rectae AB adplicare oportetj F non 
maior figura in dimidia j4B deaeripta simili defectui, 
ea autem, cui similem figuram deficere oportet, sit iJ. 
oportet igitur datae rectae j4B datae figurae recti- 
lineae T aequale parallelogrammum adplicare defioiens 
figura parallelogramma aimili figurae ^. 

secetur enim AB in duas partes aequales in 
puniJto E, et in EB describatur figurae i^ similis et 



1) Uerba a Theono lin._ G intecpolata ideo parum neceBaaria 
eunt, qaod za diSoficvov tvS-oyQafiiiov ad t^ So9ivti (bc. liSei) 
lin. 5 referri non poeBtmt, Bod neceBaario a quouis lectore ad 
Tu So&ivri tv^vyedjiUjBj Ijn. 2 trahnntaT. 

2) Hunc Siopuifiov statim praebet prop. 27. — Carapa- 
nam VI, 37; „quod eecandnm eiusdem eanm eeae parallelo- 
grammo super dinidiani datae lineae collocato minime moinB 
eiiatat" non intellego, aidetur tamen potina cnm 



corr. AngQBtaa. 6. lo Scl fooi' nttfa^ai.iiv] add. Theon 

(BPVp); m. ree. P. wopo3t«Ufiv FV. 7. dvayettqionhov] P; 
«HpKlJaUottEVoii Theon (BFVp). oitaiov] P; hixoliov ovxaiv 
Theon (BFVp), P m. ree. tra lUe^^ftoTi] P; tiBv iXlti^- 

(ioi(ov Theon (BFVp), P m. rec. 8. tov ts — 9; iXUiictiv'] 
add. Theon (BFVp); m. reo. P, 12. o»] om. P. tou] 

xa <f. tiii AB] P; om. Theon (BPVp). 13, avayQafO' 

y.ivov'] P; ntt^a^aUoiiitov Theon [BFVp). oftoCou rp iUU^- 
ftari] P[ ouolav oVTmv tav iXXtiuattxav Theon (BPVp). 18. 
TO E] euan. F. 
_ 11* 



164 



rroixEiaN p'. 

x«( eviiiiinXiiQtaU&Ba 



1 

DVOg S^9 



XEifievov t6 EBZH. 
nuQalX^loyQafifi-ov, 

El [liv ovv teov iotl t6 AH zm F, yEyovoq 
slfj t6 int-Tax&iv TiaQa^ipiriTai yaQ naffa rijv 8o- 
5 ^Etgav sv&Btav Tr/v AB ta> do&ivri ev9vy0<xfifia ta 
r iaov jcagalXiiloypa^fioi' tb AH iXXelnov Etdei 
itaQaXltjXoyQafifioi ta HB b(io(e) ovti r^ ^. si 
Si ov, litt^ov ^ffTCJ t6 @E ToiJ r. teov di to &E 
Tc5 HB' fitt^ov Rp« xal t6 HB tou F. a Sii (lettov 

10 iari t6 HB tov r, tavty ttj vneqoxjt fsov, Tro St 
jd Siioiov xal ofioiMg xeijisvov t6 avtb avvsataTa 
t6 KAMN. aXXa tb ^ rra HB [iBtiv] o^olov 3«4- 
t6 KM &Qa rra HB iativ o[>.oiov. letfa ovv 6{ 
Xoyos 71 (liv KA t^ HE. ij di AM t^ HZ. 

15 inel taov ietl rb HB totg F, KM, (itt^ov aga ii 
t6 HB TOii KM- ii£itfav wpK ietl xal ^ (ikv HE 
Tijg KA i, Si HZ r^g AM. xeie^ca tjj [ihv KA 
feij t; ffiT, rfj S\ AM tatj ^ ffO, xal GvfnrexXri- 
(fme&a t6 SHOn aaQaXl*jX6yQaii[ioV taov ccQa xal 

20 opoioV ^Srt [ro HJ7] Tto KM [aXla r6 fi^M tw HB 
"(lOiov iativ]. xal To HII K$a ta HB ofiou.v ieriV 
%£qX triv avriiv aga Sitxfi,{tQ6v iati ti HH ra HB, 
fijTOJ avriav diafietpog tj HHB, xal xatttysypa^Qai 
To exvc-t^- 



ie^ 






1. EBZH] BEZH P? 2. Post naeaU»)idyea(iuo* i 
Theoa: lo Sfi AH ijxoi laov hrl rm T jj fifffoi' auioij 
roi' SiOQiaftov (BVp, F mg. m. 1; pro SioQiaiiov habent FV 
betaftov; ia V eorr. m. 2), 3. i<niv P; in F com i6 ^H 

euan. 6. ^H] euan. F. 8. a^] 3" F. t<n<a] PF; 

forai Bpi lari V. fli rd ] a^ roi B. 9. t(5] 10 B. HB] 
H aupra m. 1 V. Sij] Se uel ietB; Jer p. 12. Ichv] 

om. P. 18. KM] inter A: et M una litt. («?) enan. F. 14. 
KA] AK Bp, 15. HB] e corr. m. 1 p. 17. KA"] AK Bp. 



ELEMENTOEUM UBER VI. 165 

similiter posita EBZH [prop. XVni], et expleatur 
parallelogrammum jiH. iam si AH »» F, effectum 
erit propositum. nam datae rectae AB datae figurae 
rectilineae T aequale parallelogrammum adpUcatum 
est AH deficiens figura parallelogramma HB simili 
figurae J. sin minus, sit @E> F.^) sed ©£ = HB. 
itaque HB > F. iam excessui, quo maius est HB 
figura JT, aequale et parallelogrammo ^d simile et simi- 
liter positum idem construatur KAMN [prop. XXV]. 
sed ^ r^ HB. quare etiam KMf^HB [prop. XXI]. 
iam correspondeant inter se KA, HE et AM, HZ. 
et quoniam HB = F + KM, erit HB > KM. quare 
etiam HE > KA, HZ > AM.^) ponatur ff fl? = KA 
et HO = ^M, et expleatur parallelogrammum SHOH. 
itaque aequale et simile^) est parallelogrammo KM. 
quare etiam HH r^ HB [prop. XXI, cfr. lin. 13]. 
itaque HH^ HB circum eandem diametrum posita 
sunt [prop. XXVI]. sit eorum diametrus HHB, et 
describatur figura [p. 161 not. 1]. 



1) Ex hypothesi; quare debnit esse iatai lin. 8, sed iatm 
ferri posse negare non ausim. 

2) Nam per prop. 20 erit HB : KM ^^ HE^ : KA^ =- HZ» 
: AM\ iam cum HB > KM, erit HE» > JT^*, HZ» > AM^, 
h. e. HE > KAy HZ > AM, 

3) Quia HB r^ KMy erit iOH!3!=^ KAM, itaque HH, 
KM aequiangula sunt. quare et similia sunt (def. 1) et aequa- 
lia (prop. 14). cfr. p. 144, 11. 



T^ lihv KA] Bp; Tij KA iisv PF; (ihv z^ KA V. 18. HO] 
corr. ex H0 m. rec. P; O e corr. m. 2 V; HB F? 20. ro 
ifJT] om. P. TflS] e corr. P. aXXa t6 JTM %m HB Sfioiov 
ioTiv^ t6 HU. dXXa t6 KM tm HB ofioiov icti supra m. 
rec. F. AM] A in ras. m. 2 V. 21. im BVp. iati, 

BPV, comp. p. 



166 STOIXEIiiN S', 

dntl ovv teov ietl t6 BH Totg r, KM, nv th 
HIl ra KM ietLv tuov, ^otjros apa 6 TX0 yua- 
(iiav Xotna tm T teos iotiv. xal iTtel taov istl xo 
OP ta SS, xoivov nfioax€ia&(a to /7B* oAoi/ «pa 
5 to OB oAra rra S!B taov iaxiv. aXka tb SB zS TE 
ianv taov, iatl xal aXevQct 17 yiE zXsvQa trj EB 
iaztv tar{' xal t6 TE «pK ta OB iativ taov. xoivoi' 
3tQ0iSxEia&-a to S£' oAov aga tb TS of.m xst *^X'T 
yvaiiovi ietiv taov. dll' o ^XT yvcofuav t<5 F 

10 idEix&fj taog- xal xb TH apa za F iattv taov. 

77apK r^v do&eiaav Spa tv&elav tijv AB rp 
So&ivti svd^vypaiifia rra F teov 3ta^aXlriX6yQttji(tov 
jEap«(3i^ATjTai tb £T iXXetaov etSii jrapaAAijAo^paft^ 
Tp IIB bfioLa ovti TM jJ [ixeid^^xep xb TIB xa HH 

16 ofiotoV iaTiv\ onep edei notiieai. ^M 

IlaQa tiiv So&eZaav ev&eiav ta do-fr^wtt 
ev&vyQa{i(>.q> taov 7ta^aXXrjX6yQa{i(i.ov attffa- 
paXetv vneQ^ttXXov stdei jrapaAAijAoypa^fip 
20 ofto^ro rp So»ivtL. 

"Eata rj ftev SoQ^etea ev^eCa ij AB. to d% SoQ^iv 
ev&vyQaft^ov , ro Sel taov naQa tijv AB itaQa^aXetv, 
tb r, ro Se Sel oiioiov vaeg^aXXeLv , xb A' Stt d^ 
naQtt. tijv AB ev&Btav ta F ei&vyQd^tfip fOov icaQaX- 
, 25 XijloyQaniiov «aQa^aXetv vneg^aXXov etSei «aQaXX^Xot^ 
yQii(i(i<3 ofLoim tp A. ^M 



1, BH] in rftfl. m. 2 V; HB p. 2. ftro» ieriv p. lof 

wov P, coiT. m. rec. TZ*] T*X P V. 3. iany taos P. 

fmiv] iariY, comp. p. iati] iaxlv P. 6. OB] euan. F. 

6. ToDv (•jtCh B. Ante ixtC add. tp: ^nV. 7. OB] O ia 



V ELEMENTOKUM LIBEE VI. 167 

iam quoniam B//= T + KM, quorum Hn=KM, 
erit etiam TX^ = F. et quouiam 0P= SS [I, 43], 
commune adiiciatur IIB. itaque OB = SB. aed 
SB = TE, quouiam AE = EB [prop. I]. quare 
etiam TE = OB. commune adiiciatur 32!. itaque 
T£=^XT. sed demoEatratum est, esse ^XT=r. 
quare etiam T£ = F. 

Ergo datae rectae AB datae flgurae rectilineae 

(r aequale parallologrammum adplicatum eat HT de- 
£cieiis figura parallelogramma I2B, quae figurae ^ 
mmilis est'); quod oportebat fieri. 
XXIX. 
Datae rectae datae figurae rectiliueae aequale 
parallelogrammum adplicare excedeus figura parallelo- 
gramma aimili datae. 

8it data recta AB, data autem £gura rectilinea, 
cui aequalera figuram rectae AB adplicare oportet, 
sit F^ea autem, cui similem figuram excedere oportet, 
sit jJ. oportet igitur rectae ^B figurae rectilineae 
r aequale paraUelogrammum adplicare excedens figura 
parallelogramma simili i 



K^' 



1) Nam JTB ~ HB (proy. 24) r^ .^. uerba inctdif xip - 
ubi Bine ca,asa de /f/7 meiitio iniicitar, spuria BQnt. 
r*eH est p. 170, 7. 



raa. m. 2 V. 8. TS] TB corr. ei TF m. 1 p. i>. al).a Bp. 
10. TS] An P. 11. nett] om. F. 13. Supra ZT ras. 

est in V, 11. tui] (tert.) postea insert. m. 1 F. 18. x*'] 
ly' p et F, corr. m. rec. 18. na^uilltnlDypaftftcv] naQalX-tilo- 
Bustulit resarcioatio in F. 22. iei] Sit Fp. 23. vnaj^alsiv F. 
Sti aij] Buetnlit lac. pergaroeui F. 34. jraeti — iv&v/iap,^ia] 
mg. m. 1 F. foov] in ras, F. 



168 



STOrXEUlN 5". 



TtTfnjaQGt Ti AB SC%ix. xttTK ro E, xal avciyE- 
Y^atpQto uno t^g EB Tp z/ ofioiov xctl ofioCas ■ 
(iBvov n«Qai.At}l6yQa(tfiov lo BZ, xal avvafttpoT^Qois 
fihv tot^ BZ, r Haov, xa 6i ^ ojtotov xal ofioCas 
6 xtCiiivov ro KUTo evvteziira rh H&. otioXoyos Si 
/ffTO ij fiiv K@ rfj ZA, 17 Sl KH t^ ZE. xttl 
insi {lEiifiv iexi xb H& tov ZB, fitC^tov aQa iatl 
xal ij (tlv K& Tijs ZA, ij Ss KHttjs ZE. ix^E^ki- 
a&aeav aC ZA, ZE, xal tj; filv K& fff»j ^utoj 

10 ZAM, r^ Si KH tari ri_ZEN, xal ffyft^KBAi/praOfiw 1 
To MN' To MN aqa rp H& laov xl iexi xaX o/to(OV. \ 
aXka To H® %m EA iaxiv ojtoiov xal t6 MN Rpa 
T^ EA ofioiov iaxiv it£Ql t^v avxiiv aQcc SiafietQOV 
iext ro EA xm MN. rn&a avxav SiafiSTQOs V ZS-, 

16 xal xttTayeyQd<p9a ro aj^^[ia. 

'ETtBlJeov ietl ro H& rofs EA, T, allu ro j 
rm MN iaov iexCv, xal to MN ap« rofff EA, |J 
leov ierCv. ^eotvov Rqa^jpjjo&o) ro EA- kotxog «f^ 
o ^X(P yvcop.(ov tm F iertv teog. xal iml toif ierlv 

ao t] -^E rfi EB, teov ierl xal ro AN xa NB, rovr^tfn 
T^ AO. xoivbv itQoaxtCe&a tb ES' oAov «pa 



3. BZj corr. ei HZ 
corr. p;^ HZ, T Y. z 

Sltoiov uQa {arl ro if© t( 
EZ P. 8. Jtei 0K F.' 

11. teI om. V. {.m.' P. 
E,l] .^ P. laziv oitoiov T 

1*. lott] sapra F. 
id] (prins) t(o F. 
comp. p. EA 



stlv 
iett 

1 



a. 2 V. 4. BZ, ri Z et r* 

e corr. F. 6. «8] PF; H»' 

ZB Bp, V mg. m. 2. 6. ZEl 

10. KH] corr. ei JTB m. rec. P. 

12. id ] {alt.) zm F, aed oorr. 18. 

Itxiv] P, corap. pi {axi BFV. 

aviav] ttvTmv i\ V. 16. iTcsl ovv PV. 

17. 1«; P,BV, comp. p, 18. hxC BV, 

mutat. ■ " " ■ " 



; comp, p. 



21. AO] Oe 






;ELEMBNT0RUM UBER VI. 169 

secetiur j4B ixi daas partes aequales in puncto E, 
et in EB figurae ^ simile et similiter positum con- 
struatur parallelogramniuui BZ, et BZ -\- F magoi- 




tudini aequale, parallelogrammo ^ autem simile et 
similiter positum idem conatruatur H@ [prop. XXV]. 
correspondeant >) autem K&, ZA et KH, ZE. et 
quoniam H&>ZB, erit etiam XS^Z^ et KH>Z£ 
[p. 165 not. 2]. produoantur ZA, ZE, et sit 

ZAM= K&, ZEN= KH, 
et ezpleatur parallelogrammum MN. itaque MN et 
aequale et simile est parallelogrammo H& [p. 165 
not 3]. sed H& ^ EA. quare etiam MN^^BA 
[prop. XXI]. itaque circum eandem diametrum posita 
sunt EA, MN [prop. XXVI]. ducatur eorom dia- 
metrus ZS, et describatur fignra. 

jam quoniam H& -= EA + r et H& = MN, erit 
etiam MN ■= EA -\- F. subtraliatur, quod commune 
est, EA. itaqae est WX^ = F. et quoniam AE — EB, 
erit AN = NB = AO [1, 43]. commune adiiciatnr 

1) Sc. in 6H, Ea paraUelograiniiiia, quae figorae ^ Bimi- 
lia snat; uude etiam inter se siinilia aont (pnip. 21). 




170 ETOIXEIflN S'. 

AS teov datl tm ^XW yvdfiovi. «kka o ""<&S 
yvdfimv rp F ffloj iariv xal ro AS aga tp P £aov 
iczCv. 

ITaQa tijv do&^ttSav aga sii&stav t^v AB t» 

5 do&evTi BV^YQaftfia rt5 /* ieov aaQaXKijloypaftfiov 

Ttaga^ipltirai ro ^S vxtp^aXXov Ei^Sti nagaXXijXo- 

y^afifia ta IIO b[io(^ ovzt zp ^, iael xal rp EA 

i6tiv ofiotov t6 OII' oasp idsi xoiijaai. 

X'. 
10 T-^v So&EtOav £v&stav iceTtEQaC^svqv Kxpov 

xal ftieov Xoyov Tt^sVv. 

"Eara r] So&ttaa svffiia asatpaenevrj 7) JB' Sil 

Sii tiiv AB ev9ttttv axgov xaX ^icov Xoyov refteiv. 

'AvayBYQatp&a ano tijs AB rezQaYOivov to BF, 

16 Jtai napa^e^X^S&a) naQa ri]V AFra BP roov itaQ- 

ttXXiiXoygaiifiov to r^t vzeQpalXov etSei rro AJ 

ofioia rip BF. 

TstQayiavov di iort xo BT' zerQaymvov affu iorl 
xal To A^. xttl ixsl tisov iotl rti BF ta TA, 
20 KOivbv tt(priQi^e&co rb TE' Xombv «pa rb BZ Aoiffra 
rra AA ietiv !Oov. iezi, cSi avta xai ieoyatviov 
rmv BZ, AA apa avtimJtov&aetv at nXevffal ai 
wspi T«s tetts yavittg' leziv «pa a>g tj ZE itQos t^V 
E^, ouiros ^ AE srpog T^r EB. i'ar] dh ij [lev ZE 
25 rfj AB, 71 Ji E^ rfi AE. eOztv aga ws ^ BA Xffbs 

1. all' F. 2. fooe] foov g> (non F). iotiv} F, comp. 
iisii PBV. 8. ieti B. 4. Si/o] aopra comp. F. ev*Bi_. 

iaxt F. 7. Ttti] (alt.) t<f F. et V, eorr. m. 2. 9. W'] p; F, 
sed corr, ni. rec. 11. rsfttiV] snpra scr. »> m. 1 F, 

yop dito PV. PoHt AB cBB. magna F. 16. ..ir] coi 

/JB m. 1 F. 20. BZ] corc. ex BT m. 1 p. 21. tm] 






ELEMENTORUM LIBER VL 



171 



ES. itaque Ag = 9XW. sed 9XW=r. quare 
etiam AS = JT. 

Ergo datae rectae jiB datae figurae rectilineae F 
aequale adplicatum est parallelogrammum AS exce- 
dens figura parallelogramma II O, quae similis est 
figurae ^, quia OII ~ EA [prop. XXIV] ; quod opor- 
tebat fieri. 

XXX. 

Datam rectam terminatam secundum rationem 
extremam ac mediam secare. 

Sit data recta terminata ^iB. opor- 
tet igitur rectam AB secundum extre- 
mam ac mediam rationem secare. 

describatur enim in jiB quadratum 
BF, et rectae AF adplicetur paralle- 
logrammum F^d quadrato BF aequale 
et excedens figura ji^ simili figurae 
BF [prop. XXIX]. quadratum autem est BF] itaque 
etiam ^i^ quadratum est. et quoniam BF = Fd^ 
subtrahatur, quod commune est, FE. quare BZ = Ajd. 
uerum etiam aequiangulum ei esi^) quare in paral- 
lelogrammis BZ^ AA latera aequales angulos com- 
prehendentia in contraria proportione sunt [prop. XIV]. 
itaque ZE : EA = AE : EB. sed ZE = AB^) et 




1) Nam utrmnque rectangulum est. 

2) Nam ZE = AT (I, 34) et AV =- AB. 



(non P). Hcov iotlv F. 23. ti}v] om. BPp. 24. AE] 

AB tp. njv] om. BFp. ZE t^ AF, rovtiatt t^ AB 

Theon (BFVp). 25. AE] AB tp. 



172 



ETOIXEIiiN S', 



t^v AE, oiJTOs 7j jiE repog i^v EB. ^Bi^av S\ ^ 
AB r^g AE' {ibl^ov aQa xccl ^ AE Tijs £B. 

'1/ eJpa j^B £w#if« axqov xa,l [leeov /oyov i 
T^ijTffi XRTa t6 E, xal t6 ftft^oi' «vT^g TiA.rifia i 
6 t6 ,/i£" o«£p ^'5et JFot^ffat. 

'£v Totg op&oyiaviotg tQiycovoig t6 k;e6 t^n 
T^v opd'i]v ycJV^av t sroritrotiflijs nXsvQag eldti 
teov isrl Tolg KJEo tav rijv oq&^^v yavlav xty^ 

10 Qiixoveav ^XsvQav eCdeSi Totg ofioiotg te xn 
ofiOiMs dvayQaq>o!i.BvoLg. 

"Eara tQiyavov 6p9oytnviov to ABF op^ij* /j(i 
r^v uroo BAF ytoviaV Xtya, ori ro areo t^g . 
£1*^05 rffou ^etl roFg Kffo rtSw B^^, AF etSeOL 

16 6jto^o(s T£ xttl oftoiog dvBygaq>0(i^voi.s- 

"Hx&a xtt&ttog ij A^. 

'EitsX ovv iv oQ&oytt^via x^iymva ta ABT ttXo 

r^g XQog Tf» A op&^s yavlag i^tl ty\v BF ^deiv 

XK^fTOg rjxtai rj AA, ta ABjd, AAV xqo^ rfj xa- 

EO 9ita tpiyava ofioia iett t^ ts oAp ta ABF x«l 
aKXriKoig. xal insX o^otov iert xo ABF ta j4BA, 
iOrtv aQtt tSg ^ FB iiQog tijv BA, ovtag ^ A3 
apog v^v B/1. x«i i^XBi r^tlg sv&Eittt avaXoySv t 
etv, Sartv ag ij xgdrjj nQog tijv rQitt]v, ovttas i 

25 «:;r6 r^g XQ(or7}s etSog ^Qog t6 Kjr6 r^g Sevtipag 1 
o^oiov xttl O(ioias ttvaypttq^onavov. ag «(>« 



XXXJ. Proclus p. «e, 14.. 



4. Mcta] tta p, xaL 10] xai' p. Imo' F, comp. p. 

10] 1] P. Sequitur alia demooBtratio , u, app, 6. la'] i 
liqnet in F; om. p. 10, emeiv PB. m] ( ""' 




ELEMENTORUM LIBER VI. 173 

Ejd = AE. itaque BA : AE '^^^ AE : EB. sed 
AB > AE. quare etiam [V, 14] AE > EB. 

Ergo recta AB secundum extremam ac mediam 
rationem secta est in E [def. 3], et maior eius pars 
est AE] quod oportebat fieri. 

XXXI. 

In triangulis rectangulis figura descripta in latere 
sub recto angulo subtendenti aequalis est figuris in 
lateribus rectum angulum comprehendentibus similibus 
et similiter descriptis. 

Sit triangulus rectangulus ABF angulum BAF 
rectum habens. dico, figuram in BF descriptam 
aequalem esse figuris in BA, AF similibus et simi- 
liter descriptis. 

ducatur perpendicularis A^. iam quoniam in 

triangulo rectangulo ABF ab 
angulo recto ad A posito ad 
basim AFperpendicularis ducta 
est A^, trianguli AB^, A^F 
ad perpendicularem positi et 
toti ABF et inter se similes 
sunt [prop. Vlll]. et quoniam 
ABr r^ ABA, erit [def. 1] FB : BA = AB: B^. 
et quoniam tres rectae proportionales sunt, erit ut 
prima ad tertiam, ita figura in prima descripta ad 
figuram in secunda similem et similiter descriptam 

13. vno x6 p. 14. sldsaiv P. 16. oiioioas'^ ofioioig V. 
18. Tffii] To FV, sed corr. m. 2. 19. AJF] corr. ex. AJBm, 
rec. P. ccQu TtQog V. 20. iauv P. 25. ro] (alt.) om. F; 
inser. m. 2, sed enan. 




174 ETOIXEIGN S'. 

n^bg Tt]v B^, ourog rb rko tijs fB «doe Bpos 
To Rjro T^? BA ZQ ofioiof xtei. oftotus avayQatp6(ii- 
vov. dia Tcc uv%a Sij xal ag ^ BF Xffbg tijv rj, 
ovtag to dab tijg BF elSog jrpog to anb t% FjI. 

p^S BF E^rfos jrpog t« airo Tt5v 5.^, AF ta oftoca 
xai oiioiag dvaygaipofttva. i9rj Sl ii BF Tatg B^, 
^r- Hsov aga xal to bWo t^j BF tlSog rotg axb 
tmv BA, AV EtSeat toZg b(Loiotg te tial ofio^o? avtt- 

10 ypa^ofi^votg. 

'Ev aga tofs op^oj^iDvtot? Tptj^dvois ro a«o 
T^v opdiji' yovCav i)not£Lvovtfi\g xXevQag etSog tOi 
ietl Totg ttrto taiv r^v dpfr^v yaviav JtEQiB%Qvamv 
aAsvpav eiSeei rotg ofioioig ts aal ofioiag uvaysfaqio- 

j5 lidvoig' ortfp sSbi Sti^ai. 

'Eav Svo rgiyava ovvte&^ xara ^lav ; 
vtav tas Svo nkevgag tatg SvoX nKiVQatg a 
loyov i^ovta wGtb tag 6(ioi6yovg avTiav JcJisrK 
eo 9^S xftl nagaXi^^^lovs slvat, at komal 
ytnvav TtksvQal in tv&siag sOovtai. 

Eata Svo tgiyava ta ABF, jJPE lag Svo stAei: 
(fag tag BA^ AT taig Svel nktv^aig ratg ^F, jdm 
avakoyov i^ovta, mg filv tjjv AB ngog t^w AS 
26 ourojs rijv ^F irpog tijv ^JE, naQakk-qXov Sh • 

S. iivayeagiafiEvov] -yp- in ras, ip. 4. ro ana t^s ^A — fl 
ilSoi jrpofi] om. p. 5. BiJ, iJT] JB, i/T y. ^ 6. ««■ 

r^s ip. 9. 5.1^] j* e oorr, m. 3 T. eMeoii' P. «vffyptif . 
f((foe (flio) P. 11. iv aeo] in ras. pOBt ras. 3 Uti ™- • I. 
Tpiymvoic] om. p. 13. foit] lai': qi. 14. etBtaiv Jl 

Seqnitur alia demoBatratio, u. app. 16. Ifi' F p. 



i 



f ELEMENTOROI LIBEE VI 175 

[prop. S3X coroll,]. quai-e ut rB : S^, ita figura 
in FB descripta ad figuram m B^ siniileiQ et Bimi- 
liter descriptam. eademdeeauaa erit etiam ut BV.r^, 
ita figura in Bfdescripta ad figuram iu FA descrip- 
tam.^J quare etiam ut Br-. B^ -\- ^F, ita figura in 
Br descripta ad figuras in B^ et ^F aimiles et 
similiter deacriptas.*) sed Br= BJ-{- ^F. itaque 
etiam figura in BT deacripta aequalis est figuris iu 
BA, AF similibas et similiter deseriptis.^) 

Ergo iu triangulis rectangulis figura descripta in 
latere aub recto angulo subtendenti aequalia eat figu- 
riB in lateribuB rectum augulum comprehendentibus 
BimilibuB et similiter descriptis; quod oportebat fieri. 

XXXII. 

Si duo trianguli duo latera duobua lateribus propor- 
tionalia liabentea iu uno angulo coniunguntur, ita ut 
correapondentia latera etiam parallela siut, reliqua 
latera triangulorum in eadem recta ertmt poaita. 

Sint duo trianguli ABF, ^TE duo latera BA, AT 
duobua lateribuB AF, AE proportionalia babentes, 
ita ut sit AB : Ar= JF: ^B, et AB parallelum 



1) Nam ABVr^ AJT. itaqae BT: FA ^TA: Pd. 

2) Sint figurae in BF, AF, AB deBcriptae a, b, c. demoii- 
etranimuB BT : BJ -^ a : C, BP: TJ = a ib. itaque 

BT: 0= rj : 6 = BJ : C. TJ : B J = 6 : C. 

r^ -i- BJ : BJ = b + €: C. 

rj-\-BJ:b-\-c = BJ:c = Br:a. BF: TJ -J-B.£j = o: 6-|-C. 

3) Nam Br;a — riJ-|-BJ!6-|-c = Br:6-f-C, qufire 
a = 6 + c |T, 9]. 

itfl'^ ] Tpoflrtfl'^ V, corr. m. 2. 20. lov -rpiyojroii V. 88. 
JT] rj V. ' JE] FE P. 24. AB] BA FV. AF] A 

e con. m. 2 V. 26. ovTto P. JT] e corr. m. 2 V. 



176 



ETOIXEIi^-N S'. 




fiiv AB Ttj Jr, T^f Se AF zy ^fi* 
EvQBtaq istlv 7) Br Ttj FE. 

'Ensl y&Q 7iaQa.kXr}k6s ietiv ij AB r^ -J-T, xnl 
sis avtKg tiiitbazaxev tv&tia ^ AF, at ivaKKK^ yto- 
5 vltti a[ i>jr6 BAF, AFjd toai KAAjjAatg elsiv. Sta ta 
avta Srj xal ^ imo VJE t^ xjreo AF^ [at] iazlv. 
aets xal ii vxo BAF tfj vjtb VAE iattv tat}. xorl 
izel Svo t^Cysava isti tcc ABFf AFE fiiav ■ya>v(av 
ti]v OTpos Tp A fita yavia tij irpog ta A tiSijv i%ov- 

10 ta, ntBQt Se tas taag ycavCaq ras jtXcvpag aviikoyov, 
mg tijv BA Jipog f^v AF, ovtag tijv Tjd XQoq tifv 
z/£, laoytavtov apa iatl to ABF tQfymvov rcj AVE 
TQiycavG)' tOt] apa 17 vjto ABF yavla trj •vito ^FE. 
iSaCi»}} Si xaX 7] V3tb AF^ trj vnb BAF tat}' okri 

15 Kpa r] VTib AFE dvel ratg vnb ABF, BAF fuij 
ietiv. xoivi] aQoSxeia&m ^ vitb AFB' ai aga vjco 
AFE, ATB tatg vno BAF, ATB, VBA taai sitstv. 
aU' a( v%b BAT, ABF, ATB Svalv oQ^ais taat 
BlaCv xaX al vao ATE, ATB aQa SvaXv oqS^atg tetu 

ao elaiv. JtQog Si] tivi £v&sia tfj AT xal r0 xgbs 
avrfj ariiisip ta T Svo sii^etai at BT, I E (lij ixi 
ta avtct (tiQr} 'xtifiivai tag i^pstvs ymviag tag vjib 
ATE, ATB dvelv og&ats taas aoiovatv in tv%E(ag_ 
aQa iatXv i] BT tfj TE. 

25 'Eav apa Svo tpiyaiva avi'ts^^ xatu (i(av ytaviea. 



3. jr] Arif (non F). 4. aCl mntat. in xaC l 
Ttai p. 6. BAF} "ABf' F. slvi Vp, 6, JF^] "ATA 
iaiiv fffij V. 10. di'\ comp, sapra m. 1 F. 11. SA} AM 
AT] in ras. m. rec. V, FA F. 12, ^011^* P. ■^r'ET| 

'■jTE F; FdE Bp et in raa. m, 2 V. IS. JTE va>vfil 
14. BAr] FA Bupra acr. B m. 1 F. 15. far] laiipm 

V m. 1, corap. p; ('iTi) iaii BP; fow flai* V m. 2. 17. bT 




ELEMENTORUM LIBER VI. 177 

lateri ^F, AF autem lateri dE 
parallelum. dico, B F et FE in eadem 
recta esse. 

nam quoniam AB rectae JF 
parallela est, et in eas incidit recta 
AFj alterni anguli BAF^ ATJ 
aequales sunt [I, 29]. eadem de causa etiam 

L rJE = ATJ, quare etiam L BAF = FdE, 
et quoniam duo trianguli sunt ABFy ^FE unum 
aDgulum, qui ad A positus est, uni angulo, qui ad J 
positus est, aequalem habentes et latera aequales 
angulos comprehendentia proportionalia, 

BAiAr^rj: JE, erit A ABF 
triangulo JFE aequiangulus [pfop. VI]. quare 

L ABr=jrE. 

sed demonstratum est, esse etiam L AF/I = BAF. 
quare erit L AFE = ABF •}- BAF: communis 
adiiciatur L AFB. itaque 

AFE + AFB = BAF + AFB + FBA. 
uerum BAF -{' ABF -}- AFB duobus rectis aequales 
sunt. quare etiam AFE + AFB duobus rectis aequa- 
les sunt. itaque ad rectam AF ei punctum eius F 
duae rectae J5r, FE non ad eandem partem positae 
angulos deinceps positos AFE, AFB duobus rectis 
aequales efficiunt; itaque BF et FE in eadem recta 
sunt [I, 14]. 

Ergo si duo trianguli duo latera duobus lateribus 



P; B"A'rF\ FAB BVp. AFB] ABT P. TBA] supra 
scr. F; ATB P. 18. aXX' ut — 19: Mv] om. P. ABF] 
AFB V. AFB] FBA V. 19. BlaC BVp. 20. bIoIB^, 

Enolides, edd. Heiberg et Menge. II. 12 



178 



STOISEI£iN : 



lag 6vo zKbvqu^ rats Svcl Tiltvpati; avdXoyov ^j^ovttt 
mOts T(ls ofioloyovg avxcov aitvpug xal xaQttA!Li}liovs 
tivai, tti kontal 1(01' zQiyfovatv aKev^fal in tv&Eias 
' onEQ idti dtt^ai. 



5 Xy . 

'Ev Tots i'eoig «vxKoi.g al ymviai. zov 
rov $j[ov(Si Xoyov ratg xe^ifpEQeiaig^ i<p' tSn^ 
^s^tjxaaiv, idv n TtQog rotg xivtQotg iav wj 
npog tatg niQiiptQsiats <»«' ^e^ijxurai. 
10 "Evtaoav i'aoi xvxXoi of ABF. ^EZ, xal ■. 
(liv TOfs xivTpoig amav rofs H, & ymvCai ieTistin 
at vw BHF, E@Z, srpos Ss tale nspLtpeQeiaig j 
iTto BAF, E^Z' kiya, ozi isrlv tog i) B F jtsgig>eQei 
«pog tijv EZ jcspi^ipsiav , oSrras ^' zs vno BB 
16 yavia «^og rijv vjto E&Z xal 17 vxo BAV : 
ziiv iino E^Z. 

KsCa^aeav yaq z^ ((Xv BF msgtipipBitt teat 1 
10 si^g oiSaidrjstotovv at PK, KJ, tfi Sl EZ TteQi^ 
(psQsia i'aat oaaidtjnotovv al ZM, MN, xkI ixs^ei 
20 ^foaav al HK, HA, &M, &N. 

'EnsX ovv i'0at sialv at BF, FK, KA jtsQi<piQEim 

XXXm, Cfr. ZenodoniB ap. Theon. in Ptolem. p 

6. Iff' p et F, corr. m. 1 
in, 9 V. 
ifi npDE loig v.hvtQoiq] mg. m. rec, P. 9. mmv FB. ^eA 
*vlot] poet lioc uocabulam cdd. Theon; hi Si Nttl of c. 
BtE {oTti F) 5ipoe lois v.ivtgaig svvtinBjievot (avvtaxafiivt 
(BFVp), Pm. rec, 12. BUr] iitt, HV in raB. F, E©Z] J 
in taa. m. 1 B, 16, Post EJZ add. Theon: xol ht (Irt 

comp, p) a HBSr (in ras. m, 2 V, HBZF P et seq. ras. T 
zancvt xeos tov @EnZ (in ras. m. 3 V) lOfieo (BFVft* 



ELEMENTORUM LDBER VI. 179 

aequalia habentes in uno angulo coniunguntur, ita 
ut correspondentia latera etiam parallela sint^ reliqua 
latera triangulorum in eadem recta erunt posita; quod 
erat demonstrandum. 

xxxm. 

In circulis aequalibus anguli eandem habent ra- 
tionem quam arcus^ in quibus consistunt; siue ad 
centra siue ad ambitus positi suni^) 

Sint aequales circuli jiBF, ^JEZ, et ad centra 
eorum H, ® positi sint anguli BHFf EGZ^ ad ambitus 




autem BAF, EJZ. dico, esse 

arc. Br : arc. EZ = L BHF : E@Z = BAF: E^Z. 

ponantur enim deinceps arcui BF aequales quot- 
libet arcus FKj KA, arcui autem EZ quotlibet aequa- 
les ZM, MN, et ducantur HK, HJ, GM, SN. 

iam quoniam arcus BF = FK = Kji, erit etiam 



1) De interpolationibus Theonis lin. 9 et lin. 16 ofir. p. 183 
not. 1; om. Gampanus YI, 32. 



m. rec. P. 21. taai] %a£ P, corr. m. rec. Blciv] om. p. 

rx] "K'r P. 

12» 



180 



ETOIXEiaN S'. 



"1 

K, KHJ^ 



ul^lctig, fffKt Bial xttl tti vao BHF, FHK, 
yaviai aKk^kai.^' loanXaaiav Kpa iaziv tj RA TCeffi- 
ipiifBLa t^s BV, ToaavraTilaeiav iazl xal ij vx!> BHA 
yiovia Ti}s ujro BHF. dia za avTct dij xal oeanla- 

B oiav iarlv 71 NE nB^tipi(/sia r^g EZ. ToeavrttTiiaeimv 
ierl xttl ij vno N&E yavia r^s ino E&Z. el apa 
fffij ioxlv 1} BA itigicpiQtLa r^ EN jtEgiipEQEiu , fa^ 
ietl xal yavia 5j vno BHA r^ vno E®N, xaX il 
ftai^av ietlv rj BA nsQiipegBia tijg EN aeQupEQtiac, 

ft£(gof iatl xal 5) VJLO BHA ymvia zijg vxo E&N, 
jtni £[ ilaeaiav, ildaoav. tieeagav 6i} Svtiov fiByc- 
Qav, diio H£V TtEQLtpEQtiav Tav BV, EZ, Svo 6h ya- 
viav rav vm BHF, E@Z, etkrinTai t^j [ilv BF 
XEQttfEQELttg xal t^g vao BHT yaviag iedxLg noMa- 

5 nXaeiatv ^ ts BA TttpKpigEitt xttl i; vjto BHA ya- 
vCa, T^g 5i EZ itegig>Egsitts Xttl Trjg vxo E&Z ya- 
vCag ^ TE EN nEQKpigEta xal ij i-nh E&N ymvia. 
xal didsLXTtti, uti ft vaEpiy^EL tj BA nsQKpiQsitt riii 
EN ^EQLq/B(f8iag, v7iEgB%Ei xaX t] vni BHA ytovia 

Tijg vno E&N ymviag, xal eC teti, /'ffij, xal eC iidegav, 
ikdeami. Hattv a^tt, ag rj BF naffnpipEia itQog Ttjv 
EZ, 0VTa>s ij vno BHF ymvia Jipog r^v vno E@Z. 
dli' ag tj vno BHF yavia ngog t^^v u;ro E&Z, 
owTog rj iwo BAF repog tjjv vao EAZ- itiTtXaeia 

6 ydg ixaTiga ixttTi^as. xaiag «pa^ BF nEQKpiQEia ngo. 
rijv EZ TtEQtfpigEtav, avTots tj te vito BHT yan/i 
npbg tiiv vnb E&Z xal r} vnb BAF npbg 
vni EdZ. 



tialv PBF. 
l 6. i,ni 

as, P. 10. iazCv P. 



1 



EeZ] E@Z BFp. 8. itiriv P. 



ELEMENTORUM LIBER VI. 181 

L BHr = FHK = KHji [III, 27]. itaque quoties 
multiplex est Bji arcus BF, toties multiplex est etiam 
L BHA anguli BHF. eadem de causa quoties multi- 
plex est NE arcus EZ^ toties multiplex est etiam 
L NSE anguli E@Z, iam si BA = EN^ erit etiam 
L BHA = EeN, et si BA > EN, erit etiam 
L BHA > E@N, et si JB^ < ENy erit 

/. JBff^ < E@N 
ergo datis quattuor magnitudinibus, duobus arcubus 
BF^ EZ et duobus angulis BHr^ E@Z, sumpti sunt 
arcus BF et anguli BHF aeque multiplices arcus 
BA et angulus BHAy arcus autem EZ et ai^li 
-E®Z arcus EN et angulus E@N et demonstratum 
est, si arcus BA arcum EN superet, etiam- L BHA 
angulum E@N superare, et si aequalis sit, aequalem 
esse, et si minor, minorem. itaque [V def. 5] erit 
arc. Br : arc. EZ = L BHF : E@Z. sed 

LBHr:E@Z = LBAr:EJZ [V, 15]; 
nam uterque utroque duplo maior est [111, 20]. quare 
etiam 
arc. Br : arc. EZ = L BHr:E@Z = BAT : EAZ. 



11. iXdtxmv iXdtxcav F. 12. (iiv'\ snpra F. Si] snpra F. 
13. EGZ] GEZ F. 17. ymv^oc] add. m. 2 F. 20. ytoviag] 
P; om. Theon (BFVp). iXdcttoiv F. 21. iXdaamv] comp. F. 
ri] om. V. 22. BHF] F add. m. 2 V. 24. amXaaitov V. 

26. yap iativ Bp. 27. v«6 E®Z] EGZ P. v«o] v- 

supra m. 1 P. 



182 STorxEKiN sr'. 

'Ev aqa totg t6ois xvxXoig at yfovCav tbv avtov 
ij^ov6L koyov tatg nBqifpsqBCavg^ iq)' mv fie^xaifvv^ 
idv te icqoQ totg xivtqotg iav ts JtQog tatg nBQupe- 
QsCaig cofft fiefitjTcvtaL* onsif idec det^aL, 

1. 'Ev] inter s et v ras. 1 litt. Y; i seq. ras. 2 litt. F. 

2. psprjnaai p. 3. iav tb — 4: psprixvtaC] %al vu i^g p. 

3. %ivtQOig'\ xvKkoig B. tag nsQiq>SQsCag Y. 4. mctv B. Li 

fine libri EvxXsCdov axoixslmv ?' PB, Ev%XsC8ov azoi%sCmv 

rTJg Simvog iyidoasmg S" F. 



ELEMENTORUM LIBER VI. 183 

Ergo in circulis aequalibus anguli eandem habent 
rationem quam arcus^ in quibus consistunt^ siue ad 
centra siue ad ambitus positi sunt; quod erat demon- 
strandum.^) 



1) Sequitnr additamentum Theonis in BFYp, de qno^ipse 
profitetur comm. in Ptolemaeum I p. 201 ed. Halma » p. 50 
ed. Basil.; om. P m. 1 (add. manus recens in mg.) et Gam- 
panus; huc pertinent etiam additamenta p. 178, 9 et 16. 
demonstratio u. in app. 



r. 

a\ Movdg ifStvv^ xa-O*' tji/ €xa6tov tmv ovtmv 
hf kiyBtai. 

/J'. 'jiifLd^fils dl to ix fiovddov 6vyx6viisvov 
6 TtXijd^og, 

y\ MiQog i6tlv aQid^fiog dQi^fiov 6 ikd66(ov rotJ 
fisi^ovog^ otav xata(i£tQfj roi' fisi^ova. 

d\ MiQTj di, otav ft^ xatafistQtj. 

s\ noXkankd6Log 8\ b fisi^cov tov ikd66ovogj 
10 orcti; xata(istQrjtaL vjco rot; ikd66ovog. 

sr'. "AQtLog aQLd^fiog i6tLV 6 8Cxa ^vaiQOviiBvog. 

g'. nsQi66og 8% 6 fiij 8iaLQOvfisvog 8l%a r^ [6] 
(iovd8L 8Laq)iQ(x)v aQtiov aQLd^iiov. 

ri . ^AQtLaxLg aQtLog aQLd^^og i6tLV b vno 
15 aQXLOv aQLd^^ov ^stQov^svog xata aQtLOV aQLd^iiov. 

d''. ^AQtLaxLg 8i 7CEQL666g i6tLV 6 V7tb aQtCov 
aQLd^iiov (istQovfisvog xatd 7tSQL66bv aQvd^^ov. 



Def. 3—6: Psellus p. 7. 6—7: Martianus Capella VII, 748. 
8. lamblichus in Nicom. p. 27. Philop. in Nicom. ed. Hoche 
1864 p. 16. 9. lamblichus p. 31. 



1. oqol] om. PB. numeros om. codd. 2. iati PBFp. 

ijf] o BFV. 10. iXdttovog V. 12. 6] om. P. 14. nQoa- 
vnanovatiov fiovov P mg. m. 1. 16. iativ'] dQid-fios iativ P, 
iativ aQL&fios p. 'nocvtavd^a nQoavnayiovatsov * (lovov mg. m. 1 P. 
tov ccQtCov deleto tov V. 



m 

Definitiones. 

1. Unitas est ea^ secundum quam unaquaeque res 
una nominatur. 

2. Numerus autem est multitudo ex unitatibus 
composita. 

3. Pars est minor numerus maioris, ubi maiorem 
metitur. 

4. Partes autem, nbi non metitur. 

5. Multiplex autem maior minoris, ubi minor eum 
metitur. 

6. Par numerus est^ qui in duas partes aequales 
diuiditur. 

7. Impar autem^ qui in duas partes aequales non 
diuiditur, siue qui unitate differt a pari numero. 

8. Pariter par est numerus, quem par numerus 
secundum parem numerum metitur.^)' 

9. Pariter autem impar est, quem par numerus 
secundum imparem numerum metitur.^) 



1) Def. 8 scriptor nescio quis, qui Philoponi commen- 
tarimn in Nicomachnm retractauit, apud Hoche Philop. 1865 
p. Y in quibusdam dvziYQcitpoig ita inuenit ezpressam: dQ- 
xidmg aQTi^og iativ aQi^^og 6 vno aQxCov dQi^yLOv %axcc &qxiov 
aQiS^fiov (lovcag fistQov^svog^ de qua scriptura falsa u. Studien 
p. 200. 

2) De def. i' interpolata u. Studien p. 198 sq. ; om. ed 
Basil. et Gregorius. 



186 STOIXEIiiN J'. 

[l\ n€QL66dxLg aQtLog i6riv o vno neQi66oi 
aQcd^^ov [i€rQOV(iBvog xarcc aQrtov aQid^iiov^ 

ia\ n€QL66dxLg dh 7C€QL66og aQLd^fiog i6%LV 
vTto X€ql66ov aQLd^fiov fi€rQovfi€Vog xara 7t€Qi666v 

6 aQLd^flOV. 

fc/J'. IlQ&rog aQLd^fiog i6rLV 6 fiovdSL (aovji ^lB' 
rpoiJftfi/og. 

Ly\ IlQSroL TtQog dkki^kovg aQLd^iioi €l6iv ot 

^OvddL ^OVrj ^€rQOV^€VOL XOLV^ ^irQG), 

10 lS\ Uvvd^erog aQLd^fiog i6rLV 6 dQLd^fip rm 
fi€rQOV(i€vog. 

L€\ Uvvd^eroL Sh JtQog dXki]kovg aQLd^fioi €i6LV 

Ot aQLd'^ rLVL ^€rQOVll€VOL XOLV^ ^irQC). 

L^\ ^AQLd^fiog dQLd^fiov 7toXXa7tXa6Ldi€LV kiy€' 
15 rat, orai/, o6aL €l6lv iv avr^ ^ovdSeg^ ro6avxdxLg 
6vvr€d7J 6 7tokXa7tka6La^6(i€Vog^ xal yivrjrat XLg. 

L^\ "Orav Si Svo aQLd^fiol 7tokXa7tka6id6avr€g akkri'- 
kovg 7toico6L rLva^ 6 y€v6fievog iTtcTteSog xaketrai^ 
TtkevQal S\ avrov ot 7tokka7tka6Ld6avreg dkki^kovg 

20 aQLd^llOL, 

Lti , ^'Orav S\ rQ€tg dQLd^^ol 7tokka7tka6Ld6avr€g 
dkkrjkovg 7totco6L rtva^ 6 yevofievog 6reQe6g i6rLVj 
7tk€VQal Sh avrov ot 7tokka7tka6Ld6avr€g dkkiqkovg 

aQLd^flOL, 



12. lambliclius p. 42. Martianus Capella VII, 751. Philop. 
in anal. post. fol. 15^. 13. Alexander Apbrod. in anal. pr. 
fol. 87. Martianns Capella VII, 751. Philop. in anal. post. 
fol. 15 ▼. 14. Philop. in anal. post. fol. 15 ▼. 16—17. Psellua 
p. 6. 18—20. Psellus p. 7. 



1. dh aQtios Pf litt. agr- in ras. aprtog ciQtd^fios p. nQoa- 
lovarsov' xa^ nata aQtiov mg. m. 1 P. 3. aQid^iiog] 



vnwKOvatsov 



ELEMENTORUM LIBER m 187 

10. Impariter autem impar numerus est^ quem 
impar *numerus secundum imparem numerum me- 
titur. 

11. Primus numerus est^ quem unitas sola me- 
titur. 

12. Primi inter se numeri sunt^ quos unitas sola 
eommunis mensura metitur. 

13. Gompositus numerus est; quem numerus ali- 
quis metitur. 

14. Gompositi inter se numeri sunt^ quos numerus 
aliquis communis mensura metitur. 

15. Numerus numerum multiplicare dicitur, ubi 
quot sunt in eo unitateS; toties componitur numerus 
multiplicatus, et oritur aliquis numerus. 

16. Ubi autem duo numeri inter se multiplicantes 
numerum aliquem efficiunt, numerus inde ortus planus 
uocatur, latera autem eius numeri inter se multi- 
plicantes. 

17. Ubi autem tres numeri inter se multiplicantes 
numerum aliquem efficiunt, numerus inde ortus soli- 
dus est; latera autem eius numeri inter se multi- 
plicantes. 

18. Quadratus numerus est aequaliter aequalis, 
siue qui duobus aequalibus numeris comprehen- 
ditur. 



om. V. 8. Sl TtQos P. 14. noXvnXaauiisiv PBp. 16. 

jtoXXaTtXaaia^o^svos] -tofisvog e corr. m. 2 p. 18. noimaiv 

PB. 22. notmaiv B. iaziv] F, comp. p; iatt P, Psellus; 
naXsixai BV. 23. Supra ot in P m. rec. Svo. 



--- ^ ^ - 



188 STOIXEKiN J'. 

cd''. Tstffdyovos AQi^^iiog i6tiv b l6aKig tffog 
rj [6] vjto Svo t6oiv aQid^iiciv nBqu%6iuvog, 

yC. Kvfiog dl 6 l6axLg l6og l6a7ug i^ [^] "^^o 

tqimv t6(ov aQi^fimv 7t6QLex6(isvog. 

^ xa\ ^AQid^fiol avakoy6v b16iv^ otav 6 Jtgditog 

tov SevtiQov xal 6 tgCtog tov tstagtov i6uxig 5 

7tokka7tkd6Log rj to avto (liifog rj ta avta (ii(fri m6iv, 

xfi\ 'X)fioiOL inCnsSoi xal 6tSQSol aqt^yi^oC 
sl6iv ol avdkoyov l^ovtsg tag nksvQdg. 
10 xy\ TiksLog aQt^(L6g i6tLv b tolg savtov fLiQS- 
6lv t6og Sv. 



a . 



^vo aQLd^ficiv dvC6{ov ixxsLfiivcJv^ piv^- 
vq)aLQOVfiivov Sh dsl tov ikd66ovog dxb tov 

15 fisC^ovog^ idv ksLn6(isvog (iriSiTtots xara^iS' 
tQtj tbv TtQO savtovj sa)g ov ksLq^d^r} [lovdgj 
ot ii, dQ%f^g dQLd^fiol jtQcitOL JtQog dkkrkovg 
i6ovtaL. 

^vo yaQ [dvC6(ov] dQLd^^icov tciv AB^ JTz/ dv^- 

20 vq)aiQOV(iivov dsl trov ikd66ovog dnb rov (isC^ovog 
ksL7c6(isvog (irjSsjtots xatafistQsCtc} tbv n^b savtoVf 
scag ov ksLg^d^ij [lovdg' kiyo^ otL of AB^ F^ TtQcitoL 
TtQbg dkki^kovg sl6Cv^ tovti6tLv otL tovg AB^ F/l 
(lovdg (i6vrj (istQst. 

26 Ei y^Q (^'^ ^i^^'^ oC AB^ r^ TtQcitoL TtQbg dkkri- 
kovg^ (istQT^6SL tLg avtovg dQLd^^i^g. (istQsCto^ xal 



23. Martianus Capella VII, 753. 



2. 6] om. PB. 3. 6] om. P. 4. tatovl om. P; mg. 

m. 1 V, supra m. 2 B; hab. Psellus, Fp. aQi&itmv tamv P. 

6. Ante ladmg in F add. ^; idem V supra scr. m. 1. 10. 



ELEMENTORUM LEBER m 189 

19. Gubus autem est aequaliter aequalis aequaliter^ 
siue qui tribus aequalibus numeris comprehenditur. 

20. Numeri proportionales sunt, ubi primus se- 
cundi et tertius quarti aut aeque multiplex est aut 
eadem pars aut eaedem partes. 

21. Similes numeri plani et solidi sunt^ qui latera 
proportionalia habent. 

22. Perfectus numerus est^ qui partibus suis aequa- 
lis est 

I. 

Datis duobus numeris inaequalibus et minore sem- 
per uicissim a maiore subtracto, si reliquus nunquam 
proxime antecedentem metitur^ donec relinquitur uni- 
tas, numeri ab initio dati primi erunt inter se. 

Nam duorum numerorum AB, Fjd 
minore semper uicissim a maiore subtracto 
reliquus ne metiatur unquam proxime 
antecedentem ; donec relinquitur unitas. 
dico, numeros ABy FJ inter se primos 
esse, hoc est, unitatem solam numeros 
ABy FA metiri. 

nam si AB, FA inter se primi non erunt; 
aliquis numerus eos metietur. metiatur et sit E, et FA 



aavTov] avxoig V, corr. in avxov m. 2. 12. «'] om. V. 

18. dvo] P; iav dvo Theon (BFVp). JxxatftcWv] ix- 

eras. P. dvd^vtpaiQOfiivov V; corr. m. 2. 14. Si] P; om. 

Theon (BFVp). 15. idv] P; om. Theon (BFVp). Post 

Ismoiisvog ras. 2 litt. V. 16. Xrjq^d'^ V. 19. dv^amv] om. P. 
tmv] Tci5 F, y add. m. 2. dv&vq^aiQO^ivov F. 21. ngo] su- 
pra m. 2 V. 22. Xriq)^y V. 23. slat Vp. 26. dQiJ&fiog av- 
tovg F. fifT^ijTco P, corr. m. rec. 



A. 


L® 


z- -r 




-H 


E 


B- 


L _^ _ 



190 



ETOIXEUiN C'. 




' xal 6 fiev r^ ri)v BZ pfTQiov ^etxh 
' Hdeoovtt xhv ZA, o tf^ AZ x6v ^H fitT^mv 
Itmitco EKVtov iXKUaovK roj' HF, 6 6i HF tav Z(t 
(iftgcov Xcmiro} ^ovaSa r^v ®A. 

5 ''EtceI ovv 6 E zov r^ fiiZQst, o di F^ rov j 
(istpEt, xkI o E aga roi' BZ fiEtgit- [lEtQeV dh i 
oAov Tov BA' xal lotnhv apa tov AZ (itrgijaEi. 
6 Si AZ Tov ^H fittQEt' xal 6 E tt^a thv AH 
fLEtQSL- ftttpEi Sh xal oAov rov ^F' xal XotTtov i 

} rhv rn (lETQijatt. 6 Sh rn zhv Z& fiEtQEi:- xaia^ 
E aga thv Z& (lEtptf ficipfC di xal oAov i 
xal loinijv apa rijv A& fiovada (iCTpjjffct aQi&ftbi 
av oJTfp ietlv ttSvvatov. ovx aqa toig AB, FA 
ttgt9ftovs (ifTpijVfi Tig aQid-fios' ol AB, F^ Kpa 



'tm 



^vo ttpi&fi.av dod^ivtcjv fii] JiQtatav «fS 
ttll-^Xovs t6 jiiytStov avtiav xoivbv (idvi/m 

EVQEtV. 

> "Eatateav ot So&ivtEg Svo api^^ol (i^ XQm 
jTpoff «AAijioiig of AB, VA. SEt Sif twi' AB, 
To ftiyiatov xoivoi' (litQov tijQEtv. 



1. BZIPP; AB BVp, P ni. re 
2. ^if] PF; ^r BVp, Pm. reo. 

3, Hri ra ?, wr] th p. 

io ras. V, P m. rec, F m. 3. 
BZ] ZB P. 6, BZ] ZB P. 7, 
UQu\ anpra comp. F. td»] to \ 

ILtTQtC^ (prins) PF; UETp^OK BVp 
To p. ^rj TiJ P. 10. TOI/] 

11. fiiijir] (prinfl) aupra nj. 2 



.; yp. t 


vAB Fmg. 


m.« 




^rmg. m. 




ZH| 




flt .* 


5. ra 


^r V ia ra 








Pp. 


^.re^o^ 6 E V. 


***! 






•M 


TO p. 


Bfionflet 


i i 




■^ 



ELEMENTORUM LIBER V7I. 191 

mimerum BZ metiens relinquat') se ipso minorem 
Z^, AZ autem nunierum ^H metiens se ipso mino- 
rem relinquat HT, HV antem numeriim Z& metiens 
reliaquat unitatem &.4. 

iam quouiam E metitur r^, et F^ metitur BZ, 
etiam E metitur BZ, uemm etiam totum BA meti- 
tor; quare etiam reliquum AZ metietur. sed AZ 
metitur ^H. quare etiam E metitur AH. uernm 
etiam totum JF metitur. quare etiam reliquum FH 
metietur. sed FH metitur Z®. quare etiam E meti- 
tur Z@. uerum etiam totum ZA metitur. quare 
etiam quae relinquitur, unitatem A& metietur, cum 
ipse numerus sit; quod fieri non potest. itaque nou 
metietur uumeros AB, FJ numerus aliquis, ergo 
AB. FA inter se primi sunt; quod erat demon- 
strandum.^) 

II. 
Datts duobus numeris non inter se primia maxi- 

I mam mensuram commuaem inuenire. 

I> Sint duo numeri dati uon primi inter se AB, r^d. 

|.iq)ortet igitur numerorum AB, FA maximam men- 

lnnram communem inuenire. 

1) Sc. ei AB. neqne enim dubitari poteBt, qaia BZ in 
F et optiiDO TheomDOTum eerDatam iiera ait acriptura, cum 
(ntfeCv eemper apud Euclidem aiguificet: aiue leaiduo metiri, 
cfr. lin. 5, 8. eajem ettt tatio lin. 2—3 et p. 19S, 11 sq. 

S) Reticui in libris VII— IX flgara.a codd., id qaod ipsa 
TSs Buadere uidebatur, uelut atatim ratio prop, t; c&m ii, qui 
pro lineia pnncta aubetitount, et in alias direcuItateB incurrunt 
et ad cerloi Dumeroe confugere coguutur, qaod ab Buctide 
alieni 



POBt priuB r^ add. V: nat t'( 



192 STorxEiiiN t'. 



11 

H€TQe€ di iud 

iutiv. 



Ei iilv ovv a r/J Tov AB [lETQtt, fiETpeC 
lavTov, r^d uQa xmv F^J, AB koivov (idr^ov iutiv. 
x(tl ^ttvt^ov, oTt xrtl (iiyiotov ovdcig ya^ fieitoiv 
Tov r^ rov r^ ftiT^fjaei. 
6 El ds ov fi6TQ6t 6 r^ rov AB, zmv AB, 
RV&v^Ki^ovjtdvov ael tov ii,aaaovos amb xov fiei 
vog lfiq>&^eiTai xtg aQiitfios, og jtaTpiju« rov srpD 
ittvrov. fiovccg fihv yag ot ^eifp&^^aeTar i! de (iij, 
i'aovTai ot AB, fz/ itQaToi tiqos a},Xr}kovs' owtp oix 

10 vnoxetxai. Ae((jp#ij<Tcrai xis aga aQi9fiai, og nexQi^aei 
xov rcpo icuroti. Kal b fs.lv Tt^ rov BE (lEtQav 
A«3ier(o iavxov iXaeaova xov EA, b Ss EA xbv JZ 
fiet^av keiTciTto iavtov ildasova rov ZF, 6 Si FZ 
tbv AE fieTifeixm. iael ovv b FZ rov AE [leTQtl, 

16 6 de AE rov AZ jiexQet, xal 6 PZ apa rbv 
(iBTQiqaef (iBTQei: dh x^l savxoV xal olov apa 
r^ fierp^tfei. 6 S\ FjJ rov BE fietQBf xal b 
KpK tov BE fterpef' (iBTQft Se xal xbv EA' ««l SJ 
opK Toi' BA (iBXSfriasi,' (tetQst ds xal tov F^' b 

ao aQa roii' AB, VA (lex^ei:. 6 fZ «pa xav AB 
xoivbv fiitQov iativ. kiyia S-ij, ort xal (tiytetw. 
ei yctQ (iij eaxiv 6 VZ xtnv AB, fz/ (iiyiaxov xoivbv 
{letQov, (lexQ^aei xts tous AB^ r.d ttQt&(iovs a(fi 
fibs fieit^ov ^v xov TZ. (iexQeixoj, xal iarto 6 

25 xttl inel 6 H tov TJ fteTQet:, 6 Si TJ xbv BE 

a. rj, AB} AB, rj P. l<nt. BFV; corap. p. f 

8' e. 6. a{fi Theon (BFYp). (latrovos FV, 1 

if&^eetaiVn, con:. m. 1, 8. iliiqj&^OETat p; P, corr. m 

10. Iriip&iiafTai p. «?«] aupra m. 1 F. Squ ns V. 
supra m. 1 F; mg. m. reo. B. II. B£] PF; AB 

Pm. rec, yp. lo» AB mg. m. 1 F. 12. .JZ] PF; r^ 
jr B, V in raa. m. 2, P m, recj toi' JP F r 



!Qti, 

] 



ELEMENTORUM LIBBB VU. 



193 



iz 



iam si F^ metitur .^B, et etiam se ipsum metitur, 
r^ communis erit meusura numerorum F^, AB. et 
adparet, eum etiam maximam esse. neque enim 
ttllua uumerus numero Tz/ maior metietur r^d. 

at d r^ non metitur AB, minore numerorum 
rf£, FiJ semper uicissim a maiore subtracto relin- 
quetur numerus aliquia, qui proxime aute- 
cedentem metietur. unitas enim non re- 
linquetur; sin miuua, AB, Fz/ iuter se 
primi erunt [prop. I]; quod contra iiypo- 
I ~ tlieeim est ergo uumerus aliquis relin- 
ij 1 quetur, qui proxime antecedentem meti- 
etuj-. et rj metiens BE relinquat se 
minorem EA, EA autem zJZ metiens relin- 
se ipso minorem ZT, rz autem AE meti- 
|ktur. iam quoniam PZ melitur AE, AE autem 
^^Z metitur, etiam FZ metietur AZ. uerum etiam 
se ipsura metitur. quare etiam_totum T.^ metie- 
tur, sed Tz/ metitur BE; quare etiam FZ metitur 
BE. uerum etiam EA metitur. quare etiam totum 
BA metietur. uerum etiam Fiit metitur. ergo FZ 
metitur AB, FA. itaque FZ communis est mensura 
numerorum AB, FzJ. dico iam, eum etiam maximam 
rZ numerorum AB, F^ communia 
Biensura masima non est, aliquis numerus maior 
mero FZ numeros AB, PA metietur. metiatur, 
li sit H. et quoniam H metitur J^z/, FJ autem BE 



Zr] rz BVp. dO om. B. li, Ante iwi io V i 

ii EA (in TEis. m. 2) ittvziyv ilDiraova ov ^ii^tith {xov 
rZ. 21. iiSTi BV, comp. p. 



194 STOIXEI^ J'. 

tQSt^ xal 6 H aga xov BE fierget' ^bxqbI 6% Ttal 
okov xov BA* xal koLitov aga xov AE fi6v^0Bi, 
8% AE xov dZ fiBXQBt' xal 6 H aqa xov ^Z fi^ 
XQTi^BL' iiBXQBt ds xal okov xov d F' xal komov aga 
6 xov rZ iiBXQT^6BL 6 fiBiicov xov ikd66ova' 0X6Q i6xlv 
aSvvaxov ovx aQa xovg ABj Fd aQtd^ftovg agiQ^iiog 
xtg (iBXQi^0BL iiBiicDV cov xov FZ' 6 rz aQa xAv ABj 
Fd (liyLiSxov idXL xolvov ^nixQov [omq ISbl d£r|ca]. 

n6QL(i(itt. 

10 ^Ex dij xovrov q^avBQov^ ozl iav aQLd-fiog 6vo aQid'- 
fiovg iiexQijj xal xo fiiyL6xov avrmv xolvov [litQOV 

liBXQrj^BL' OTtBQ i^BL ^Bt^aL. 



TqlAv ccQLd^iiciv dod^ivrcDV pLTi jCQcixoiv nQog 
15 dkk7]kovg t6 iiiyL6rov avrciv xolvov (lixQov 

BVQBLV. 

"E6rGi6av OL dod^ivrsg rQstg aQLd^iiol ^rj TtQmxoL 
TCQog dkl^ijlovg oC A, B, F' Set drj rciv A^ Bj F ro 
liiyL6rov xolvov fiirQOv svQStv. 
20 EiXi^^pd^ci) yaQ dvo rciv A^ B ro ^iyL6rov xolvov 
[lirQov 6 A' 6 Sri A rov F ijroL ^srQst 7] ov ^srQBt. 
lnsrQsCrca TtQorsQov ^srQst Sh xal rovg Aj B' 6 A 
ccQa rovg A^ B, F iisrQst' 6 A ccQa rciv A, B^ F 
xoLvov ^irQov i^rCv. kiycD dij, orL xal ^iyL6rov. 



3. fieTQSt' xat] corr. ex a£tQr]Gsi m. 1 p. xov /H, aga F. 
liexQi^aei.] fiszQSt P. 4. xov] corr. ex to m. 1 p. ^T^] 

rj p. 6. iaziv'] om. B. 8. iaziv PV. 10. tovxo P, 

sed corr. 12. onsQ idev dsiiai] P; om. BFVp. 19. fts- 

xQov] bis p. 20. 8vo yuQ p. 22. fiexQsi:] (alt.) om. F. 

24. iax^v] comp. Fp; iaxC PBV. ^?}] om. P. 



ELEMENTORUM LIBER Vn. 



195 



inetitur, etiam H metitur BE. uerum etiam totum 
BA metitur. quare etiam reliquum AE metietur. 
sed jiE metitur jdZ, quare etiam H metietur ^Z. 
uerum etiam totum AF metitur. quare etiam reli- 
quum rZ metietur maior minorem; quod fieri non 
potest. ergo numeros AB, FA non metietur numerus 
maior numero FZ. ergo FZ maxima est communis 
mensura numerorum ABy F^, 

Corollarium. 

Hinc manifestum est^ si numerus duos numeros 
metiatur, eum etiam maximam eorum mensuram 
communem mensurum esse.^) — quod erat demon- 
strandum. 

m. 

Datis tribus numeris non primis inter se maximam 
mensuram communem inuenire. 

Sint tres numeri dati non 

primi inter se A^ B^ F. oportet 

igitur numerorum A^ By F 

^ maximam mensuram commu- 

nem inuenire. 

sumatur enim duorum numerorum A^ B maxima 

mensura communis A [prop. II]. A igitur aut meti- 

tur r aut non metitur. prius metiatur. metitur 

autem etiam A, B, /1 igitur numeros Ay B, F meti- 



B 



E 



1) Nam H et AB, FJ et comnmnem eornm mensiiram 
maximam FZ metitur (p. 194, 5). 



13' 



196 



ETOIXEKiN £'. 




bI yap fi»; ionv 6 ^ rrav A, B. T fidyierov xotviv 
lidtgov, (letQ^^eu ns TOtr^ ^. B, F api&iiovg aQi&fibs 
liti^av av lov ^. fittQiiTa, xal iUTco 6. E. ixil 
ovv o E Tows J, B, r fifTpsi, xal tows j4, B «p« 
6 fterpijOff xal to rtav ^, B apa (liyiaTov xoivov ne- 
rpov |;t£T(>^a£i. To di zav A, B fiiyttftov xof.vov (ti- 
TQov iozXv b ^' 6 E agtt tbv iJ fisTffei 6 fisi^mv 
Tov ikdeeovtt' omg iexlv aSvvatov. ovx «pa tous 
A, B, r tt(fi&fiovs dgi&ii(ig tie jtETp^Oti fttittav av 
10 tov jJ- 6 ^ aiftt tav A, B, F ftiyieTov iati xoivov 

(liTQOV, 

Mij {j.sTgttra dri 6 ^ tbv F' kiyia aQtoTov, oit 
01 r, tJ ovx slot jTQatoi itQos ftAAijAoug, insl ya^ 
oC A, B, r oijx slat, npmtot jrpog «AA^Aovs, [istQijaii 

16 Ttg avTovs ttQt&^os- b dij toiig A, B, F (tttQiav xal 
tovg A, B ftftQ^^an, xul to tmv A, B fiiytotov xoi- 
vhv fiitQov tov ^ fiatQ^aEf netQsi 8i xal roi' F' 
TOirg i/, r ttQa uQt&fiovs aQt&fios tig fieTQijeef oi 
^, r aQu ovx siai TtQioToi Jtpog «AAjjAowg. e£i.^ip&a> 
ovv ttvttov tb fiiytatov xoivbv (litQov b E. xal eml 
6 E tbv /i fistget, b Si ^ toi's A, B iisTQel, xal b 
E aga rotig A, B ftEtQsf (istQst Sh xal tbv f' o E 
apa rous A, B, V ftsTQei' 6 E aQa tmv A, B, F "ot- 
vov ^0Tt (litQov. i.i-yat dij, ott, xal (liyiOvov. ^H 

SG yiiQ (iiq ietiv 6 E tav A, B, r ro (liyietov xo(i^^| 



. ycip] c 






2 ¥. 



Kaivo)! itsyiaTov V, 
.._.. . . - . T. E] COIT. Kt r I 

g. laiiv] om. Fp. 9, aQiS-fios] om. F; tig] om. 

(3»] om. P, !2-,f^] sapra F. 13. F, -J] ^, r BVp, 

15. api&fios Qiiiotie F. tovsj corr. ex toij m. rec. F. 17. 
Toi'] rd FV. (leipijOfi tou J p. 13. oei9(tojJs] m. 2 V; 

om, BF. dei&fios] V, at/i9iiOvs 9. 21. fiei^ct] (alt.) 



1 



ELEMENTOEUM LIBER VII. 197 

quare d commimis menaura eat numeronim 
!, r. dieo, eimdem maximam esse, nam ai /t 
r maxima mensura communia non 
est, Dumerus aliquis numero ^ maior numeros ^, B, V 
metietur. metiatur et sit E. iam quoniam E nume- 
ros ji, B, r metitur, etiam A, B metietur. quare 
etiam maximam menauram communem numerorura 
A, B metietur [prop. II coroll.]. uerum mazima men- 
aura eommunis iiumerorum v/, B est ^. itaque E 
metitur ^ maior minorem; quod fieri non potest. 
itaque numeros A, B, F non metietur numerua maior 
nnmero ^. orgo ^ masima est mensura communis 
numerorum A, B, F. 

iam ne raetiatur z/ numeruni F. dico primum, 

numeros P, -J noii esse primos iuter se. nam quo- 

B, r primi non sunt inter ae, numerus ali- 

'quis eos metietur. qui autem A, B, F metitur, etiam 

ji, B metietur, et z/ maximam menauram communem 

numerorum A, B metietur [prop. II coroll,], uerura 

etiara F metitur. quare numeros ^, F numerus ali- 

qnie metietur. itaque z/, F primi nou simt inter se. 

latur igitur eorum mauma mensura communis 

[prop. II], et quoniam E metitur ^, J autem A, B 

letitur, etiam E metitur y/, B. uerum etiam F me- 

E igitur j4, B, F metitur. quare E numero- 

, B, r communis est mensura, iam dico, eun- 

^e. nam si E numerorum j4, B, F 



bia F. Xdl D E apu tove J, B futfil] mg. m. 2 B. 23. 
r] inBert. m. i-ec. B. kowdv] bia P, sed, eorr. 24. ffij] 
om. P. 95. Tol om. p. 



198 ETOIXEIilN £'. 






HdzQov, (i£ipijo*e Tig Tovg y(, B, F a(fi&(iovg i 
fibg ii$i^(av av tov E. jieipeirii}, xal iaro 6 Z. xal 
ixsl 6 Z xotig A, B, r [itTQeT, xal Tovg ^. B (ut^bV 
xal xb rav A, B uQa fii-yietov xoivbv fiix^av (t£ 
6 tp^ff«. to S^ zmv A, B (isyi.GTov xoiviiv (lizQov 
iaxiv o ^' o Z «pa tov A (itTQtf litiQeZ dt xal 
zbv r- 6 Z apa tovg ^. F (ittQBf xal ro tiav ^, F 
KQa ^iyLarov xoivox' [lizgov (itzQjjSet. ro di twv d, 
r (liyiarov xotvbv (iszqov iatlv 6 Eb Z apa tov 
10 E (iBzpsi & fieitcov' tbv ikaaaova' onsg iatlv a&^a- 
Tov, ovx aga roug A, B, r api^fiovs agi,^(t6g tig 
ftetp^ffEi fASt^tov av tov E' b E aQtt tmv A, B, F-^ 
(iiyiatov iati xoivbv (iitgov omg idti dsl^at 



e "Axag aQi&[i6g xavTbg affi&(iov o iXdaem 
tov fisitovos ^Tot (tigog iatlv i\ (iiQi]. 

"Eaziaaav Svo agi&fiol oC A, BP, xal iatat iXdm 
Gmv o BF' Xiya, ort 6 BV ToiJ A ^toi (ligog i 

n /**'?'?■ 

Q OC A, Br yaQ ^roi jcptoToi npog aXK^^Xovq 
aXv ^ ov. iOtaaav npottgov of ^, BF Tcpmtoi : 



1, aei9(n)i7s] om. P. 4. «ea] om. V. (titp^*] am. 1 
7. idvl ti> F, sed corr. lo] aupra m. 1 P, .d, r1 e C ^ 
m. 9 V. 11. Bei&u.oij£] comri. F;^ om. Vp. 13. iatai 

Poat (i£i(io»' add. BV: Tpno* a^a npifrfitn* Si^ivxaiv lyo^, 
10 fityiinoi' 1101*0» jiEtew. hti^ai^ P; Ttoiijaai Theon (BFVp). 
8eq. in p, B ia mg. imo m. 1, V tng. m, 2: nopiafia. ik d^ 
(eraa, B) TOtiioti (tovcdiv V) ipaft^oi', oti iav agi9nog zgt£s 
dei^jtovg lif^QV, K«l 10 fifycatov avtav xoi*oi' (lereo* (itTp^- 
CEi. 6uoi'((JE ^E xqI Tclitoviov aQi&jiap 3o9tvzaiv fiij ^Qio-taiv 
xgoE all^lovi TO fiiyiaiov aOTiur (om. Vp) koivob liizifoii 
cvg^BKftat Mol ro noemiia agoiioQ^uet. Praeterea V in testu 



ELEMENTORUM LIBER VH. 199 

maxima non est mensura communis, numerus aliquis 
maior numero E numeros ^4, B, F metietur. metiatur 
et sit Z. et quoniam Z numeros ^, Bj F metitur, 
etiam ^4, B metitur; quare etiam maximam numero- 
rum ji, B mensuram communem metietur [prop. II 
coroU.]. uerum numerorum j4, B maxima mensura 
communis est ^. Z igitur z/ metitur. uerum etiam 
r metitur. Z igitur ^, F metitur. quare etiam nu- 
merorum ^, F maximam mensuram communem me- 
titur. uerum numerorum ^, F maxima mensura com- 
munis est E. Z igitur E metitur maior minorem*, 
quod fieri non potest. itaque numeros A, B, F 
non metietur numerus maior numero E. ergo E 
maxima est communis mensuii^ numerorum A, B JT; 
quod erat demonstrandum.^) 

IV. 

Minor numerus maioris semper aut pars est aut 
partes. 

Sint du^p numeri J4, BF, et minor sit BF. dico 
BF numeri A aut partem aut partes esse. 

nam A, BF aut primi sunt inter se aut non primi. 
prius Ay BF primi sint inter se. diuiso igitur BF 



1) Cfr. p. 194, 12. proprie oec dBilai nec notr{Gai, sed 
Bv^Bvv dicendum erat (Studien p. 62); nam propp. II — III no- 
Qiafiara sunt (ib. p. 61). inde consecuta est uariatio scripturae. 



habet: tov avTOv 8h tqotcov xal nXstovoDV aQid^fimv Sod^ivTav 
t6 (isytaTOv ytoivov (istqov svQrjaofisv. 15. Zinag] Zi littera 

initialis add. m. 2, ut semper fere, V; eras. B; babent Pp9. 
17. iXdTzcDv F. 18. Xiyo} ott] in ras. qp. 6 BF tov A"] 
eras. F. 21. ngoTSQOi V. ol Ay BF] mg. V. 



200 STOIXEK^N J'. 

dkkiikovg. dLaLQed^dvtog dfi xov BF eig tag iv avtf 
^ovidag i6tai ixa^tri (lovag tmv iv tp BP [liQOg tt 
tov A' &6ts ^iQfj i6tlv 6 BF tov A. 

Mri i6t{o6av dii ot A^ BF TtQcitot XQog aXkrj' 

5 Xovg' 6 8rj BF tbv A ritoi (i^Qet rj ov fiatQst. sl 
fihv ovv 6 BF tov A ^istQst, ^Qog i6tlv 6 BF 
roi; A, si dh ov, sikiiq^d^a) tAv A^ BF iiiyi6tov xoi- 
vov fiitQOv 6 Aj xal StTjQi^Gd-a) 6 BF sig tovg t^ J 
t6ovg tovg BE, EZ, ZF. xal ixsl 6 A tbv A (U- 

10 tQst, fiiQog i6tlv o A tov A* l6og 6\ o jd ijcd6ta 
tAv BE, EZ^ Zr- xal sxa6tog aQa tmv BE^ EZ^ ZT 
tov A (isQog i6tCv' &6ts fiiQrj i6tlv 6 BF tov A. 

^ATCag aQa aQtd^iibg navtog aQi^^nov 6 ikd66mv 
tov fiSL^ovog fitoy fiiQog i6tlv rj (liQtl' oitSQ ids^ 

15 dst^aL, 



s . 



^Eav dQtd^iiog dQcd^iiov (iSQog y^ xal stSQog 
itsQov tb avtb fiiQog jj, xal 6vvaiiq)6tSQog 
6vva^q)oriQOv rb avzb ^SQog i6rai^ otcsq 6 
20 slg roi ivog, 

^AQid^fibg yccQ 6 A [dQtd^iiov] rov BF iiSQog i6t(0j 

xal srsQog 6 A iriQOV rov EZ rb avrb fiiQog^ otcsq 

b A rov BF' kiyto^ on xal 6vvafiq)6rsQog b A^ A 

6vvaiitporBQOv rov BF^ EZ rb avrb fiSQog i6rlv^ otcsq 

26 A rov BF, 

'ETtal yccQj o ^SQog i6rlv 6 A rov Br^ rb avrb 



1. drj] yaQ, supra scr. drj F. sccvxip p et F (corr. g>). 
2. Tt] F; TO 9?. 4. ot A, BF] om. V. aXlriXovg ot A, BTY. 

7. zb fisyiatov BFp. S. 6 BT] F; ABF cp. ^ tc5] corr. 
ex x6 p. 9. %a£] om. BFp. 10. Ss] di^ P. B-naxsQca Yq). 

11. xof/] F; 6 qp. aga tov V. 13. iXdtToav (p. 18. 17] 

P; om. BFVp. 21. aQi&nov] om. P. fiSQog] F, fiovog 9. 



ELEMENTORUM LIBER m 



201 



A 



B- 



Et 



zJ- - 



in suas unitates unaquaeque unitas in BF compre- 

hensa pars aliqua erit numeri ^; quare 
BF numeri ^4 partes erunt. 

iam ne sint u4, BF inter se primi. 
itaque BF aut metitur j4 aut non metitur. 
iam si 5F metitur A^ pars est BF nu- 
meri ^4. sin minus^ sumatur numerorum 
j4, BF maxima mensura communis z/ 
et diuidatur BF in partes numero ^ 
aequales, BE, EZ^ ZF. et quoniam z/ metitur ^4, 
pars est ^ numeri j4. sed z/ = BE = EZ = ZF, 
quare etiam unusquisque numerorum BEj EZ, ZF 
pars est numeri j4. quare BF partes sunt numeri j4. 
Ergo minor numerus maioris semper aut pars est 
aut partes; quod erat demonstrandum. 



r- 

[prop. II], 



_B 



Si numerus numeri pars est, et alius numerus 
alius numeri eadem pars, etiam uterque utriusque 
eadem pars erit^ quae unus unius. 

nam numerus A numeri BF pars 
sit, et alius numerus z/ alius nu- 
meri EZ eadem pars sit, quae A 
numeri BF, dico, etiam j4 -^- ^ nu- 
meri BF -]- EZ eandem partem esse, 
quae sit A numeri BF. 
nam quoniam quae pars est u4 numeri B F, eadem 



-E 



T +e 

- -r - iz 



22. iiiqog] fisgog iazCv (-iv m. 2 e corr.) V. 
BF] mg. m. 2 V. 24. EZ] F, BZ 9. 

1 V. xo avTo] zovzo P. 



23. Xiyto — 25: 
26. 0] supra m. 



202 



rroKEiiiN £'. 




(idQog iozl xal 6 A %ov EZ, oOot aga eIoIv 4v t^ 
Br aifi&jiol teoi rp A^ toaomoi tiet xal iv rm EZ 
aQi&iiol teoi. Ta ^. StjjpTjoO'© 6 (itfv BV siq rovs 
xa A tsovs tovs BH, HV, 6 di EZ sig eow? rm d 
6 /ffoug Towff £0, 0Z' Effrai d(j fSov ro jrA^&og Ttov 
BH, HV TM rcAiJ^si Taiv £©, 0Z. xal ^jtsl fffos 
im\v h ftii/ BH rra A, 6 Si E& ta J, xa\ ol BH, 
E& «pK TOtg A, A iaoi. dia tii avxa 8ij xal ot 
HF^ &Z totg A, A. offot «pc [tl'ffii'] iv rw ST 

10 ftptfrftol i'6oi ta A, Toffot7to(; eisi xal iv rolit BF, 
EZ taot rofg A, ^d. o6aitXaeiav aga iariv 6 BV 
tov A, toaavtaTtXaeCiav ietl xal evvaiupotspog n 
BP, EZ avvaiiq>otiQov to€ A, A. 6 aga i^iQog ietip 
6 A TOV Br, tb avto fiiQog iati xal ewafigioTEgos 

15 b A, A avvaji^otigov tov Bf, EZ- ojrsp Idet d'ff|( 



'Eav aQi&{ios dgt&fiov fttp'? fli "«^ crjpos 
ctiQOV ta avta ftigi] tj, xkI evvafiiporsgos evv 
aftipotipov ta avta iisqti setat, onsp 6 eis 
rov Bvog. 

'Aqi&(i6s yag 6 AB (fptdftov Tou V fiiQij letco, 
xal 'itEQoq 6 ^E Etigov rou Z tq: avta (iiQ*i, ansQ 
6 AB roii P' Xiyto, ozt xal ewaiupotEQog 6 AB, AE 



Iff^ 



1. ioTiv F. v.cti] in ras. m. 2 p , insort. m. 2 F. J] 
corr. BZ A m. "i p. (ipa] fpa aQi9iio( V. 2. aei^rio^] om. v. 
A] ^ ip. daiv PB. 7. Post d add. Theon: o BH ««a tu 
A laos iini (lutiv B) (BFVp). 8, «0«! om. Tbeon (BPVp). 
ioDtj om. Theon (BFVp). za aira) xaira V. Poat ffij 

add. Theon: xoi o HF ta A fooff (F, rooi' q?) iaxiv (W ,V, 
eomp. p) (BFVp). In V praeterea add. «Bi 6 8Z rio J. 
' "" ez roft ^, jl] Hr t^ A fooe W«, o *i'<9Z 
I Hr Toie AiJ ip (nou F). In emendatione praeiait 



oE HV, 




ELEMENTORUM LIBER VII. 203 

pars est etiam jd numeri EZy quot sunt in ET 
numeri numero A aequales, totidem etiam in EZ 
numeri sunt numero A aequales. diuidatur ^JT in 
numer^ numero A aequales BHy HT^ EZ autem in 
ESy &Z numero ^ aequales. erit igitur multitudo 
numerorum BH, HT multitudini numerorum E®, @Z 
aequalis. et quoniam est BH=Ay E®=Ay erunt 
J5Jff + E® = A -^- A, eadem de causa etiam 

HT+eZ = A + ^, 
itaque quot sunt in BT numeri numero A aequales, 
totidem sunt etiam in 5F + EZ numeris A + ^ 
aequales. quare quoties multiplex est BT numeri A, 
toties multiplex est etiam BT + EZ numerorum 
A + A, itaque quae pars est A numeri BT, eadem 
pars etiam A + A sunt numerorum BT + EZ\ quod 
erat demonstraudum. 

VI. 

Si numerus numeri partes sunt, et alius numerus 
alius numeri eaedem partes, etiam uterque utriusque 
eaedem partes erunt, quae unus unius. 

Nam numerus AB partes sint numeri JT, et alius 
^E alius Z eaedem partes, quae AB numeri T 



Augustus. 9. xoig] agcc torg V. z/] d teoi elaiv V. ocoi] 
oa- in ras. m. 2 F; tarj (p (non F). siaiv] om. P. 10. 

sioLv PB. 12. iativ P. 13. o] om. cp (non P). fiigog] F, 
fiev (f, 15. Ssl^ocl] noLTJaai V. 17. fiiqog p. 21. a^t^- 
fiov] aQi&fiov €p (non F). 22. z^E] E sapra m. 1 V. 23. 
ozi avvaiKpotSQOi ot p. 



202 STOIXEISiN f. 

^dgog iotl xal 6 jd rov EZ^ o0ol aqa eiolv iv xA 
Br aQid^fiol t6oi xm A^ xo6ovxoC b16l xal iv xA EZ 
agid^iiol C0OL rS jd, dtrjQrj^d^co 6 iilv BF slg rovg 
rp A t6ovg rovg' BHj JffJT, 6 6\ EZ slg rovg r£ A 
5 l6ovg rovg E®^ ®Z' l6rai dri t6ov ro ^Xf^d^og rmv 
BHj HF rp nkri^^SL rAv E&^ @Z, xal i%sl t6os 
i6rlv 6 (ihv BH rm A^ 8\ E® rc5 A, xal ol BHj 
E@ aga rotg A^ ^ t6oi. dia ra avra 8ij Ttal 01 
HFj &Z rotg A, ^. 0601 icQa [sl6lv] iv rS BF 

10 aQLd^fiol t60L rS Aj ro6ovroL s16l xal iv rotg BF, 
EZ t60L rotg A^ A, 66a7tla6LC3v aga i6rlv 6 BF 
roi) A^ ro6avra7cka6CGiv i6rl xal 6vvaiiq)6rsQog 
BFj EZ 6wa^(poriQ0v rov A^ /1, 6 aQa [liQog itfrlv 
6 A rov Br^ ro avro ^sQog i6rl xal 6wa(iq>6rsQ0g 

l& 6 Aj A 6vva^q)oriQov rov BF^ EZ' otcsq sSsc dsHiaL, 

€\ 

'Eav ccQLd^^og dQcd^^ov ^SQfi ri^ xal srsQog 
sriQOV rd avrd ^SQrj rj^ xal 6vva^(p6rsQog 6vV' 
a^(poriQov rd avrd ^SQrj s^rac^ otisq 6 slg 
20 roiJ sv6g, 

^AQLd^^og ydQ 6 AB dQLd^^ov rov F ^iQrj i^roj 
xal srsQog b AE sriQOv roi) Z rd avrd fiSQri, ditSQ 
6 AB. rov F' XsycDj oxl xal 6vva^q)6rsQog b AB^ JS 



1. iat^v F. Tiai] in ras. m. 2 p, insert. m. 2 F. J\ 
corr. ex ^ m. 2 p. apa] dga aQtQ^fio^ V. 2. dgi^fioi] om. vL 
A] J qp. siaiv PB. 7. Post J add. Theon: 6 BH dga ta^ 
A taog iaxC (iazLv B) (BFVp). 8. ccQa] om. Theon (BFVp) - 
taov] om. Theon (BFVp). ta avtd] tavza V. Post 8^ 

add. Theon: xal 6 HF to5 A taog (F, laov qp) iatvv {iatCX^ 
comp. p) (BFVp). In V' praeterea add. xal 6 0Z r» -^ 
oi Hr,0Z toig A, J] o HV ta A taog iatCv, 6 ^s S^ 
tfp d F; 6 HT totg Ad q> (non F). In emendatione praei"^ 




204 ETOIXEiaN £'. 

6vvafiipoTipov Tov r, Z TCC KVta fldQlJ iOTlV, SlKff 

6 AB TOv r. 

'Eittl yap, a tiigri iarlv 6 AR lov F, tk avxa 
fispjj xal AE Tov Z, o'e« apa ierlv tv i:a AB 
5 fiifiy] Toii r, toffai^TK itfti xul iv Ta AE ftt^Qtj tov 
Z. SvtiQ^a&c} 6 (tlv AB sig ru row F fiiptj za AH, 
HB, 6 Sl ^E d? tK ToO Z ^{'pij iK ^@, ©£■ 
i6tm Sij taov ro ^A^&os zmv AH, HB Tta 3t^^&« 
Tmv ^@, &E. xal intC, S ^igos icrlv 6 AH tov F. 

10 to «vro fiigog ieil xal 6 ^^® toi; Z, o «pa fttpug 
iarlv o AH tov r", to avTo fiifog iOTl xal Gvva^- 
^dttpog h AH, .tJ& ewajifpoTipov tou F, Z. tfjc 
TK avTtt Sii Ttal o fiSQOs iOTlv o HB rov T, ro awto 
fiipos iovl xctl evvafupoTe^os 6 HB, ®E evvafifpoTi- 

15 pow tow F. Z. « apa ^Ep») ^Otlv d AB tou J", i« 
Kvre fifpi? ^«'ti xal avvccfitpon^oq o ^B, ^fi ffwK;!- 
qjottpou rov F, Z" onrfp l'd£( 5£f|«(. 

£'■ 
'£av aptS-fiog np(-&fiow ft^pos 37, Oi 
20 prfl^fis atpaiQs^ivTog, xaX 6 Aoiwog 

reoi' to «UTO fiipos iatai, osrfp o oAog toij oJtov. 

'AQi&fiog yap 6 ^^.8 «piftftoii tov fl4 (lipog lara, 

oirep «93a[p6'&elg 6 AE aipaiQi&ivTog tow fZ" AEyo, 

oTi xal Xainbg 6 EB Aoi^ioii roO Z^ t6 «wro fii^og 

25 iOtiv, ojrep oioi,' 6 .^B oAow roil PA. 

i. i/E] E e corr. m. 2 F. 5. ^oti] om. B. 6. JH] 

A corr. eji J P. 7. ^E] EJ p. 10. iaxiv BP. 11, 

i<iTiv'\ iaxiv Kai F, sed xai del. xai] uul 6 p, IB. 3^1 

del, m. 2 P. iaxC V. to evta fttpoff iot/] xul o EG tou 
Z- S Bp« fiEpDs ^ail. 10 HB TOu" r P. 14, x«i] xffi o p. 

1&, «] supra B. 1 V. 16, ioii:» PB, 18. £'] om, V, m 



I 

.01- II 



jA 
H 

1b - -E - 



r -d 
e 



ELEMENTORUM LIBER VH. 205 

dico, etiam AB -]- ^E numerorum F -\- Z easdem 
partes esse, quae sit AB numeri F, 

nam quoniam quae partes est AB numeri F, 
eaedem est AE numeri Z, quot sunt in ^^ partes 
numeri F, totidem etiam in ^jE sunt partes numeri 

Z. diuidatur ABin AH, HB partes 
numeri F, JE autem in A®, SE 
partes numeri Z. itaque multitudo 
numerorum AH, HB multitudini 
numerorum A®^ &E aequalis erit. 
et quoniam quae pars est AH nu^ 
meri F, eadem est etiam A& numeri Z, AH-^ ^& 
eadem pars erit numerorum F -^- Z, quae AH nu- 
meri F [prop. V]. eadem de causa etiam quae 
pars est HB numeri JT, eadem pars est HB + ®E 
numerorum F -{- Z, ergo quae partes est AB numeri 
JT, eaedem partes sunt AB -{- AE numerorum F + Z; 
quod erat demonstrandum. 

VII. 

Si numerus numeri eadem pars est, quae ablatus 
numerus ablati, etiam reliquus reliqui eadem pars 
erit, quae totus totius. 

Nam numerus AB numeri F^ eadem sit pars, 
quae ablatus numerus AE ablati FZ. dico, etiam 
reliquum EB reliqui ZA eandem esse partem, quae 
totus AB sit totius FA. 

quo haec prop. a. m. 1 solo signo : r^ a priore dirempta erat; 
corr. m. 2. 20. 6] supra m. 1 P. 21. 6] supra m. 1 P, 
om. F. olov] in ras F. 23. AE] A eras. V. 24. xa/] 
%al 6 BFVp. 25. oXog] 6 oXog B. 



.. TO ttVTO fli^ 



206 STOISEmN f. 

"O j'((p (legog iarlv t> AE xov PZ. 
Qog i6tca xkI 6 EB lov FH. xal i%ti, o iieqos 
ierlv u AE roiJ TZ, ro ai'ro [iJgog iorl xal 6 EB 
xov VH, o Kpa ^i^oq iozlv o AE tov FZ, ro avto 
6 liiQos iorl xcti 6 AB zov HZ. o d^ (UQag ierlv 6 
AE rou rz, ro «itio (tipos vnoKEfcai, xkI l. AB 
rov FA' o «pa (lipos iorl xal & AB rou HZ, ro 
avvo (itQos iazl xal zov FA' tVog «p« iozlv 6 HZ 
tM r^. xotvog dyjjpjjoS^ro 6 FZ' iootos «9« o HF 

10 Koi^a ro3 ZA iazLv Haog. xal inei, (iipog iozlv 
6 AE Tov rz. t6 awTo ^^po; [iazl] xal 6 EB zov 
Hr, fffog di b Hr rro Z^, Kp« ^fpo; iOTlv 
AE Toii rZ, z6 avzo (i^pog iazl xal 6 EB toO ZA. 
aXla o [lipos iazlv 6 AE Tot' TZ. t6 ei'ro fifpoi,- 

15 iazi xal o AB zov F/l' xal koiitog apa 6 EB koLnov 
zov Zjd zo avzo (lipos ioxCv, owfp oAos 6 AB oiou 
rou TA' ojtep BSet Sst^ai,. 



'Eav agL&iios aQL&fiov (liQij J/, aita(f 1 

so pEQ^tlg dg>aiQ£&ivzog, xal 6 Aoinos zov Xotam 

TK aiJTa (*E'ei Sazai, uxbq 6 oAoff rou oAou. 

l^ptftftos yiip 6 .<^B dpi&fiov rou T^ ^^pi) ^9* 
a%eQ dfpatQEd^elg o AE dtpaipe^evzog roS FZ' /e'j*| 



7. fot/i- PB, comp. p. HZ] coiT. et 
xai] Kal .^B Theon (BFVp); AB add. 
Pcist r,J add. Theon; JB &qk ivjtTi^ox 
«uio (i^eoe ietiv (BFVp); idora P, mg. m, 
Vp, 9. icoi*iDfi P, porr. m. 1 et insuper m. rec 
11, laxi] om, P. 12. BT] r in ras. F. 
6 Hr rco JZ P in ras. 0?«] om, F. 1 
loS Zij] AB TOv r,d F, corr. m, 2. 14. 



tmv H7., r^ ri 
rec. HZ] ZH 
10. fooff iat/ 7. 
*£] di ■«(' Vp. 
. ^ut^v P. EH 
;r P, corr. m. 1. 



ELEMENTORUM IIBER VH. 207 

nam quae pars est AE numeri FZ, eadem pars 
sit EB numeri FH. et quoniam quae pars est j^E 
numeri FZ, eadem pars est EB numeri FH, etiam 

AB numeri HZ ea- 

A K B 

I ! i dem pars est, quae 

H r z ^ AE numeri TZ 

fprop. V]. supposu- 
imus autem, AB numeri FA eandem partem esse^ quae 
sit AE numeri FZ. itaque quae pars est AB numeri 
HZy eadem idem pars est numeri FA. itaque HZ — Fzl, 
subtrahatur, qui communis est, FZ. itaque HF — Z^. 
et quoniam quae pars e^t AE numeri FZ, eadem est EB 
numeri HFy et HF '^ ZA^ quae pars est AE numeri 
TZ^ eadem est EB numeri ZA. uerum quae pars est 
AE numeri TZ^ eadem est AB numeri TA. ergo 
etiam reliquus EB reliqui ZA eadem pars est, quae 
totus- AB totius TA\ quod erat demonstrandum, 

VIIL 

Si numerus numeri partes sunt eaedem, quae abla- 
tus numerus ablati, etiam reliquus reliqui eaedem 
partes erunt, quae totus totius. 

Nam numerus AB numeri TA eaedem partes sint, 
quae ablatus AE ablati TZ, dico, etiam reliquum 



aXXa ] in ras. m. 2 F ; o apa post ras. plua quam 2 linn. V. 
^E] 'EB V; e corr. F. rzi in ras. F; Z^ V. ^ 16. Po8t 
Tj^ add. Bp: o aqa (isQog iavlv 6 EB tov Zd^ to avto (li- 
poff iatl Kttl 6 AB tov FJ; idem P mg. m. reo. nal Xomog 
aga] %aC mutat in o et in mg. add. aqa fiigog iatCv F m. 2 
{Xomog aga in init. lin. seq. (a m. 1) intactum relinquitur). 

16. iatC V. 17. rj] BT F. 21. 6] om. Pp; m. 2 F. 

22. rj] r add. m. rec. P. 



208 ETOIXEIfiK f- 

ozi xal Aoinog 6 EB Xoinov rov Zjd zec av%a ftdf^ 
iativf axBff olos 6 AB okov tov F^. 

KtieQa yttQ roi AB fffog o H@. a uqu (lip^ 
icriv 6 H& TotJ Fz/, T« ovtw ft^pij ^ffti xal 6 AE 
5 Tov rZ. dijjpijod-to (' (n^v H@ iCs TTtt Tow r!^ liipij 
TK ifJf, X0, 6 di AE sig TK Tov TZ ftipjj r« ^J, 
^E' ^atai Sii laov to stA^^og tmv HK, K& ew 
reAijff£( ■rrai' v4^, ^E. x«l ^wf^, o ftE(>og ^oriv o 
/fK roti Ji^, To avro (liQOs iotl nal o AA zov FZ. 

10 (itC^av di o Fj^ tov FZ, (lei^av Kpa xal 6 HK tov 
AA. xeio&a ta AA £eos o HM, o a^a [lEpos iatlv 
6 HK tov r^, rb avto jtdQos i6xl xaX 6 HM xov 
rZ' xttl Xoi.7i6s aga 6 MK Xoiitov Toii Z.d tb avto 
[ijQog ictiv, OTtBQ 0A05 6 HK oiou tow F^i. itahv 

15 ijtti, o (iipog istlv o K® rov F^, to «wto (idpos 
i<ltl xttl 6 EA Tov rz, ftei^av Si o FJ rov TZ, 
fif^grav «pa xal 6 @K zov EA. xdiS^o} rp EA fffos 
6 KN. a(fa fiEpos iatlv o K& roij FA, z6 avto 
(liQog ietl xttl 6 KN tou PZ' xal Xotitos apa 

20 N@ AotreoiJ TOiJ Z^ tb avtb (lipos iativ, otccq oilos 
6 K@ oiow Tou r^. ideixdT] di xfti Jlotjros o MK 
Aoiirov Toi? Z.^ To avTo (tiffog Sv, o«£p 0^05 o JfK 
o/ou Tou FjJ' xal avva(i<p6tBQ0S aga 6 MK, N8 
tov dZ ta avta (liQi] ictCv^ a%£Q o^log o @H oAw 

1. K«/] %al o V. ZJ] J add. m. 2 F. 2. Slos] 
oios B. 4. ^otn l<rxi'» r. 8, ^E] in raa. V. 9. H^I 
X poatea inaert. V, itiTiv PV. Ttai] om. P. 11. HM] 
MH Vp. 11. iaiiv PF. 16. lorf-v F. i:ov TZ] m. 2 aupra 
8ct. F. 17. ex] K9 P. 18, KN] con-, ei KH m. ree. 
p; mutet. in KH m. 3 V. 19. /lenieos Pi corr. m. S. 

ifciv F. nai loimosl loiTios V, 20. ivei corr. ex H8 

m. reo. p, Zz/] ^J erai, V. ojr*e] m, 2 V. 31. iJti- 
X9ii Si — 23: rj] mg. V. 21. «0^ kkI BFV, 6] om. p. 



ELEMENTORUM LIBER Vn. 209 

EB reliqui Z^ easdem partes esse, quae sit totus 
AB totius rj. 

ponatur enim H® = AB. itaque quae partes est 
H0 numeri Fz/, eaedem est etiam AE numeri FZ. 
diuidatur H® in HK^ K® partes numeri FJ, AE 
autem in AA, AE partes numeri TZ. itaque multi- 

r z *^ *^^^ numerorum HKj K0 mul- 

' ' ' ' titudini numerorum AA, AE 

^ ^^ ^\ f aequalis est. et quoniam quae 

A A E B pars est HK numeri Fz/, 

' ' ' ' eadem est AA numeri FZ, et 

rj > rZy erit etiam HK> AA. ponatur HM'^ AA, 
itaque quae pars^est HK numeri FA, eadem est 
ifMnumeri FZ. quare etiam reliquus MK reliqui ^z/ 
eadem pars est, quae totus HK totius Fz/ [prop. Vll]. 
rursus quoniam quae pars est K0 numeri FAj eadem 
est EA numeri TZ, et rj>rz, erit etiam &K>EA. 
ponatur KN = EA. itaque quae pars est K0 nu- 
meri Tz/, eadem est KN numeri TZ. quare etiam 
reliquus N0 reliqui ZA eadem pars est, quae totus 
K@ totius Fz/ [prop. VH]. demonstrauimus autem, 
esse etiam reliquum MK reliqui ZA eandem partem^ 
quae totus HK totius sit Fz/. quare etiam MK + N@ 
eaedem partes sunt numeri z/Z, quae totus @H totius 



22. mv] om. p, ov V. HK] KH P. 23. FJ] JT FVp. 
MK] eras. V. N9] corr. ex HG m. 2 p. 24. JZ] AZ F; 
ZJY^. 9H] He FVp. 



Enclides. cdd. Hcibprpr et Menge. II. 



-e 



210 STOIXEISN £'. 

TOti r"^. fflog 6i awaiKpoTepoq iihi o MK, JV* 
rw EB, 6 S% &H To Bv4' x«i i.oinos apa o £B 
(lotjrou Tow Z^ ra rtwti /a^pi; hr(v, ujteff olog 6 
AB Zlov tov r^' oiTf^ tdei dtl^at. j^H 

'Sav api#fiog aQt-&fiov ftf^os fl, 3«ai frepop 
ixtQov To ocvTO ^cpo^ fjj xul /vaAAa|, o fiipos 
iatlv ^' ftepij o jrptoTos tou t^Ctov, to avto 
(isQog ItStat ^ Ttt avTa f«^(>ij xa2 6 dEi^rE^o^ 

10 TOl TfTKproV. 

Wptfl^/ios yap 6 ^ a(ii#fM)v rov BF lii^og iatBi, 
xal etegog o jd Iziqov tot EZ to uvto jiifiog, ojtip 
6 A xov BT' \iyei, ozi xul ivttXla^, o fiipog iczlv 

A TOtJ J ^ ft^p')) ro auto f«/()os ietX xtiL 6 BT 
16 TOV fiZ ^ fi£'(»I- 

'fiwfi yap o [lipog ierlv o A tov BF, to ttvto 
(iipos iOzl xal 6 ^ zov EZ, oo^ot KpK Bielv iv za 
BF aQi&iiol taot za A, zoaovzoi elci xaX iv r^ EZ 
leoi 1(5 ^. StyQ^a&ta o \t,\v BF eIs Tot's ra A 
20 iaovi zovg BH, HF, 6 dh EZ eCs zoiis Tp ^J taovs 
rowg E&, &Z- iazai S^ i'aov z6 nl^&os zmv BH, 
HF r6 jtX^^&Ei t6v E®, ©Z. 

KttX ixsl laoi eialv ot BH, HF aptffftol «ilAi}- 
Aots, deX Sl xaX oC E&, @Z aQi9'(iol i'Got dlX^rjXoig, 

1. r^l Jr BF. *^] y corr. es a^; aij PBFp. ttje 

01 ojih V. MK, JV©] muUt in HM, KN m. 2 V; Joi- 
Bos tiQa a MK, JV© rtS EB foos iffTi'r mg. m. 2 7. 2, tw] 
e corr. m. 1 F, EB] BE Y m. 1, AE m, 2. ©H] ©JVp. 
B^] mutat. in MK m. rec. p. 3. Z^] corr. ex JZ m. 2 V. 
iJZ F. 6. Poet fiEpos rae. 6 litt, dem loii jipioTWJ ii.ei'Sovos 
Tov StvTtQov pmictia dsl. F; totam protasin ita, ut apnd noe 
legitur, in mg. repetit m. 2. 7. jj] P; om. BFVp. 9. 



ELE-MENTOEUM IJEER VIL 



211 



I 



r^. sed MK + N& = EB') et ©if = B^. ergo 
etiam reliquus EB reliqui Z^d eaedem partes sunt, 
quae totus ^B totJas F^; quod erat demODstrandum. 

IX. 
Si uumerus uumeri pajs est et aliua numerus alius 
numeri eadem pars, etiam permutatim, quae para uel 
partes primus est tertii, eadem para uel partes erit 
seeundus quarti, 

Nam numerus A numeri BF pars sit, et alius ^ 
alius numeri EZ eadem pars sit, quae A numeri BV. 
dico, etiam permutatim numerum BT eandem partem 
uel paxtes esae numeri EZ, quae pars uel paxtes sit 
A numeri zi. 

Nam quouiam quae pars est j4 numeri BF, eadem 
eat z/ numeri £Z, quot sunt in BF 
numeri numero A aequales, totidem 
etiam in EZ sunt numero ^ aequa- 
les. diuidatur BF m numeros BH, 
Hr numero A aequales, EZ aiitem 
iu E&, &Z numero jd aequales. ita- 
que multitudo uumerorum BH, HF 
multitudim numerorum E&, @Z aequalis est. et quo- 
niam BH= HF et E& = ©Z, et multitudo numerorum 



1) Nftm HM + MK+ KN ■{■ NB = AA + AE + EB, 
t HM= AA, KN = EA. 



foTui] iexi oomp. 

pnnctie del. fiepc, ,.. 

m. 2: lidzxtav dt ^ova o A t 
aupra o aor. Znsii m. 1 p. 
%vi] om. P. 18, ilmv PB. 
84. tUiv P. Ee] EZ p, 



PoBt latio add. Vi fl ttt «uto (jfpTj 

13. Poflt Br add. BVp, F mg. 

u d {J m ras. m. 1 B}. 3] 

14. ietiv F. 17. tniv PF. 

31. ftmn] ^fl*^ F, corr, m. 3, 



Mte 



212 ST0IXEK2N J'. 

ocai i6xiv t6ov ro nXri^og rmv BH, HF r«5 xk^^Bt 
tmv ES^ ©Z, o &Qa [iSQog i6xlv o BH %oi ES 
Sj [iBQfj, xo avxo fisQog i6xl xal 6 HF rov SZ ij 
xa avxa ^Qfj' &6xe xal o [liQog i6xlv o BH tov 
5 E& ^ [J^iQV^ ''^^ ^^^ ^iQog i6xl Tcal 6waiig>6t€Qog 
6 BF 6wa(KpoxiQOv xov EZ rj xa avtcc ft^pi^. t6o$ 
8h 6 [ihv BH rcS A^ 6 8h E& xm z/* o &Qa fidQog 
i6xlv 6 A xov j4 ri [liQrj^ xb avto [liQog i6tl xtd 
Br tov E Z fj ta avxa iiBQrj' otcbq IdsL det^at, 

10 . v\ 

^Eav aQL^iLog ccQvd^iiov [liQri g, xal stsQog 
BtBQOv ta avta [LiQrj y^ xal ivaXkai,^ a (liQfi 
i6tlv TtQmtog tov tQvtov rj [liQog^ xa avta 
[liQri i6tav xal 6 SsvtBQog tov tBtaQtov f^ to 

15 ec^ro [liQog. 

^AQi^fiog yaQ o AB aQcd^^ov xov F iibqti fyx&j 
Tcal BXBQog jdE ixiQOv xov Z xcc avxa (liQrj' Xiym^ 
oxv xal ivalla^j a (iBQrj i6rlv b AB xov jdE ^ ft£- 
Qog^ xa avxcc [liQri i6rl xal 6 F xov Z rj xo avxb 

20 iiiQog. 

^EtvbI yccQj a iiiQrj i6xlv 6 AB xov F, tcc avxa 
iiBQrj i6xl xal b /lE xov Z, o6a ccQa i6xlv iv xm 
AB liBQrj xov r, xo6airta xal iv rc5 jdE ^iQrj tov 
Z. dcriQrfid^m 6 ^bv AB Big xcc xov F ^iQrj ta AH, 

25 HB^ h 8\ AE Big xcc xov Z ^BQrj xcc A&^ &E' Scxat 

2. E0} corr. ex EZ m. 1 F. 4. &ozb'\ -zb in ras. V. 

7. 8b'\ 8ri P. 12. ^] P; om. BFVp. 13. Ante ^ in p 

del. %aL ft^po?] corr. ex fts^iy p. 14. ^atai (jLiQrj V, 

7La£] m. 2 F. IQ. AB] inter A et-B duae litt. eras. V. 

iaTon] qp, ^tftai? F. 17. Post fisQTj add. BFVp: icToo 9s 
{ds m. 2 F; iXocTTODV 9s icToa B) b AB tov JE ixdatfav (m. 



ELEMENTOBUM LlBER Vn. 213 

BH, HF multitudini numerorum E&, &Z aequalis 
eat, erit etiam HF niimeri ©Z eadem pars uel partes, 
quae BH numeri E®. quare etiam quae pars uel 
partes est BH nomeri E&, eadem pars uel partes 
est BF numeri EZ [prop. V et VI]. . sed BH = Ay 
E& •= d. ergo quae pars uel partes est A numeri 
^, eadem pars uel partes est etiam BF numeri EZ\ 
quod erat demonstraDdum. 

X. 

Si numenis numeri partes sunt, et alius numeruB 
alius numeri eaedem partes, etiam permutatim quae 
partes nel pars primus est tertii, eaedem partes uel 
pars est secundus quarti. 

Numerus enim AB numeri V partes aint, et aliua 
AE alius numeri Z eaedem partes. dieo, etiam per- 
mutatim numerum T easdem partea uel partem esse 
Z, quae AB numeri AE. 

nam quoniam quae partes est AB 

numeri r, eaedem est etiam AE nu- 

tJ meri Z, quot sunt in AB part«s nu- 

^ meri T, totidem partes uumeri Z in 

T® ^£ sunt. diuidatur AB in AH, 

HB partes numeri T, AE autem 

i-E - in A®, &E partes numeri Z. erit 



S P, om. B). 18. S] om. F. 19. iniv F. lov] om. p. 
21. o] m. 2 B. 22. agal m. 2 F. 24. rl in rae. 4 

litt e corr. P. 25. HB} H e corr. V. JE] E ia raa. P. 
J©1 J e corr. p; poat ras. 2 litt. V; AB F (sed J o corr.). 
0E] E eraa.; fuit EB P. 



212 ST0IXEK2N g'. 

ouci itfrcv t6ov xo nkri^o^ rmv BH^ HF x^ 
tmv E&y ©Z, o aQa (isQog i6tlv 6 BH 
^ ^Qfj, to avtb (isQog iotl xal 6 HF to 
ta avta fisQri' &6ts xal o ^SQog i6tlv i 
5 E@ 7] (liQriy to avtb ^iQog i6tl xal 6vpa 
o BF 6wa(jLq)0tiQ0v tov EZ fj ta avta fd 
d\ 6 [ihv BH t£ A, 6 8s E@ tm jd' o S 
i6tlv 6 A tov j4 ri (liQT^^ tb avtb ^fog 
Br tov EZ fj ta avta ^SQri' otcsq 18bi 

10 . i\ 

^Eav aQtd^^bg ccQvd^^ov ^iQrj y, xe 
stiQov ta avta ^iQrj y, xal ivakX&^ 
i6tlv 6 TtQmtog tov tQitov iq iiiQog^ 
liiQrj l6tav xal 6 8svtSQog tov tstaQ 

16 avtb (liQog, 

^AQvd^^bg yccQ 6 AB aQLd^fiov tov F 
xal stSQog o jdE stsQOv tov Z ta avta fi 
oti xal ivalkd^j a ^iQrj i6tlv b AB tov 
Qog^ ta avta [liQri i6tl xal 6 F tov Z 

20 fiiQog. 

'Eitsl yaQj a [liQri i6tlv 6 AB tov . 
liiQri i6tl xal b ^E tov Z, o<Ta aQa i 
AB iiiQrj totf r, ro6avta xal iv ta ^j 
Z. ^irjQr^^d^a) 6 ^sv AB sig ta tov T ft, 

25 HjB, h 8\ /iE sig ta tov Z ^iQrj ta A® ^ 



2. E0] corr. ex EZ m. 1 F. 4. SatB] 

7. ^6] dfi P. 12. -^] P; om. BFVp. 18. 

del. 'nai. fi^9og] corr. ex fiSQrj p. 14. ; 

%a£] m. 2 F. IQ. AB] inter A etB duae . 

iarm] qp, iaTcci7 F. 17. Post (isqti add. BF 
{Si m. 2 F; iXdtTav Ss iatm B) b AB tov dlr 



214 



STOKEIiiN J", 



d»; [«ov to wA^&os Trov ^H, HB xa itXi^&et xSv 
J&^ &E. xttl ixBi, o [i^QOs iarlv 6 AH rov V, to 
avTo (liQos istl xal 6 ^& tov Z, xkI ivaild^, 5 iiepo^ 
iSzlv 6 AH xov J@ ^ f*f'p*h TO «UTO iUgo^s iftl 
6 xai 6 r rou Z ^ rec «vtk ftif?' ^'^ ^^ «wra rf^ 
xai, o ni^os iatlv o HB toO &E tJ ft^C^, t6 avro 
ftE'()os f'0T^ xkI 6 r' tov Z ^ Tc: «Otk (iiQ^' mtttt 
xkC [o (legos istlv o AH tov ^® rj (ii&ij^ to avto 
fiBQOs iOxl xal HB rov &E tj rcc avta fidpr]' xal 

10 o «pa fiSQoe ierlv 6 AH tov ^& ^ ('■^9*1^ ^o auto 
ftfpog ietl x«l 6 v^B Toil ^E »} TO Ki^To: fii(»;* aXk' 
o nEQOs istlv o AH tov ^& ^ fii^^! t6 ttvzo fiitfos 
idsix^ ^i*' " ^ ^"^ Z ^ Ttt aura fiE^i?, x«l] « [«$«] 
fiipjj iotlv 6 AB Tov ^£; rj fiiQog, tk «ura fiepi; 

15 iatl Xttl 6 rroti Z ^ ro otjro fieQos' OTieg iSst dei^at. 



olov, ovrcog dtpat^gi 
ittl o Aoinog Xffos T&l^ 



'Eav 5j a>s o?.os Kj 
Oflff npog afpatsfs&sv 
lotxbv letai, cog oiog ;rp6s oioi'. 
"Eera rog oXog 6 ,4B rcpoj oAov rov 17^, otJri 
afjfiipt^&El^ o AE Jtpog Kipttt.QS&ivTa rov I^Z" A^] 
ort xtti Aotffos 6 EB repos Aomov ^6» Zj^ iutiv, 
oAog 6 ^S XQos oAoiJ rov r^:/. 

'£wst ietiv 6s 6 AB xffos tbv FzJ, outos 

1. ffnl Si p. ,^ir, JfB] in raa. ip. 2. J0, ®E] 

era9.F. 3. ««/] (&lt.}Pp, Em. rec,- om. FV. 4. JS] ©^E 
5. r] post rftfl. 1 litt. F. TH auitt] om. p. ^ 3ni ta — 7_: 
fiFpij] om. Vi mme xoi o HB zoi 8E «o «uio (on liseoe ij 
fi^ei;. onte 6 fffos rm HB, TOUTEOriv o AH, zm Com xia ^B, 
TovriatLv rm 9E p\ idem V mg, m, 1 bis {niiiog teTiv, tov 
6. UB] BH F. 10 ttvro fic^s] bia P, 



Uk 



^ 



■ ELEMENTOEUM LIBER vn, 215 

igitut multitudo numerorum j4H, HB multjtudini uu- 
merorum ^&, &B aequalis. et quoniam quae para est 
jlH numeri F, eadem est ^& numeri Z, permutatim 
quae pars uel partes est AH numeri ^&, eadem pars 
uel^ partes est etiam F numeri Z [prop. IX]. eadem de 
causa etiam quae pars uel partes est HB uumeri 
&E, eadem pars uel partes eet F aumeri Z. quare 
etiam quae partes uel pars est AB numeri ^E, eae- 
dem partes uel pars eat etiam f numeri Z'J; quod 
erat demonatrandum. 



XI. 

i est ut totus aii totum, ita ablatus ad ablatum, 
itiam reliquus ad reliquum erit, ut totua ad totum. 
Sit AB : FzJ = AE: FZ. dico, esse etiam 
EB:ZJ = AB:r^. 
quoniam est AB : Fz} = AE : FZ, erit AE eadem 



K«tiai 

^B 1) Nam AH eadem parB nel paiteB est numeri J8, quae 
r^SB numeri SE. eigo (prop. V et VI) AB uumeri ^E, eadem 
pars uel partes est, qnae AH numeri JS» siue quae F nmneri 
Z. — Bed quae hano ipsam ratiocinatiooem contJDeDt uerba 
lin. 8 — 13, merito anctoritate codicia P TheoQi tribuenda ease 
uideri pOBannt (Campanu^ iu Iub libris arithmeticia tanto opere 
. a Giaecia disciepat, ut peiraro ei eo documenta peti poaaint). 



i 



joon, m. a. 1. (ireos] eias, F. iaxl xttil 
p,iqoq — 13: fi^e^ %ai] mg. m. rec, P. 8, S a. 
aqa itipoe Vp, HS toO — 11: »«l 6] 



13. apa] m. ree.P. 14. lativl ^oti koiVij. 15. iffrft- P. 
p,t7. (d;] om, p. !S. □ Zoihdc o Y. Poat n^oe add. Y: olo» 
■» rj wpos tov, deL m. 1. ZJ] JZ P. 24. Poat iwtf 
'. yrie FV m. 2, P m. rec. 6] (alt.) in ras. m. 1 B. 



216 STOIXEKiN f. 

JtQog xov rz, o ccQa fieQog i6xXv o AB zw T^ ^ 
yi^iqtl^ xo avxo fiSQog i6xl xal b AE xov JTZ ij xa 
avxa iiiffi» xal Xomog aga 6 EB Xoixov xov Z^ x6 
avxo fiiQog ioxlv ^ i^^iQ^j ansQ b AB xov JT^. b6xiv 
^ aQa (bg 6 EB nQog xbv ZA^ ovxGng b AB XQog xbv 
FA' onsQ idsL dst^ai. 

'Eav (O0LV bjto6oiovv aQid^fiol avdXoyov^ 
S6xat d>g slg xmv fiyovfiivov TtQbg sva xmv 

10 iTCOfiivcov, ovxcog anavxsg ot iiyov^svov JtQog 
anavxag xovg s^ofiivovg. 

"E6x(o6av bno6oiovv aQi^^ol avdkoyov oC A^ B, 
r, A, ag b A nQbg xbv B^ ovxa}g b F TtQbg xbv A' 
liyoa, oxv i6xlv fhg b A TtQcg xbv B, ovxcag ot A^ F 

15 jtQbg xovg B^ A. 

^EtcsX yccQ i6xiv mg b A n^bg xbv B^ ovxcog 6 F 
TtQbg xbv A^ o uQa ^SQog i6xlv 6 A xov B rj fiiQrj^ 
xb avxb ^iQog i6xl xal 6 F xov A 7] ^iQrj. xal 6w- 
a^q)6xsQog ccQa b A, F 6vvaiiq)oxiQOv xov B^ jd xb 

20 airtb [liQog i6xlv ij xcc avxcc ^iQfj^ cinsQ 6 A xov B. 
S6XLV ccQa (bg A TCQbg xbv B^ ovxcjg ot Aj F JtQog 
xovg By A' oJtsQ sdst dst^aL, 

vy. 

^Eccv xi66aQsg dQvd^iiol ccvcckoyov o6vv^ xal 
25 ivaXkal^ avakoyov s6ovxai. 



Xin. Philop. in anal. post. fol. 18. 



1 Tov] om. V. 2. iaxLV F. 3. Xoinog] Xomov p. Zz/] 
z/Z P. 4. anso] -nsQ eras. F. 6] bis p. 12. dvdXoyov^ 
om. Vp, euan. F. 13. 6 F] 6i tp, o F — 14: E, ovto>$' 



ELEMENTOBUM LIBER Vn. 



217 



A 
E 



Zr 



BJ- z/J- 



pars uel partes numeri FZ, quae AB nu- 
meri T^ [def. 20]. quare etiam reliquus EB 
reliqui Zz/ eadem pars uel partes erit^ quae 
AB numeri TA [prop. VII. VIII]. ergo 

EB.ZA^ AB iTJ 
[def. 20]; quod erat demonstrandum. 

XII. 

Si quotlibet numeri ptoportionales sunt, erunt^ ut 
unus praecedentium ad unum sequentium^ ita omnes 
praecedentes ad omnes sequentes. 

Sint quotlibet numeri proportio- 
nales A^ B, T^ Ay ita ut sit 
AxB^T.A. 
d dico, esse A :B = ^ + r:B + ^. 
nam quoniam est ^ : 5 = F : ^, 
quae pars uel partes est A numeri 
B, eadem pars uel partes est etiam 
T numeri A [def. 20]. quare etiam A -^- T eadem 
pars uel partes sunt numerorum B + ^, quae A 
numeri B [prop. V. VI]. ergo 

A:B = A + T:B + A [def. 20]; 
quod erat demonstrandum. 

xm. 

Si quattuor numeri proportionales sunt, etiam 
permutatim proportionales erunt. 

om. p. 16. A'] in ras. m. rec. P. x6v\ x6 9. 17. 0] 
^ 9 (non F). Tov] tov qp. 19. 0] e corr. V, m. 2 P. 

20. htCv'] comp. F, enan. Dein in F seq. 23 folia perga< 
meni recentissimi (97); incip. ^<7Tly 17 xtX., desin. IX, 15 fin.: 
det|at. 21. PoBt lcTTtv in £: o, del. m. 2. 24. co<a Vp^. 



JB 



218 STOIXEiaN s', 



o^ JTSr^T 



"Esxaeav tiaeaffEg d^i,%yiol avdXoyov ol - 
z/, tos o j4 apog rov S, outojg 6 F jrpoff zoi' ^" Xdyco, 
0T( xa^ ^aAAug Kvciloyov ieovTtti, as < -^ ff^og roi' 
r", oftrms 6 B jrpog rov ^. 
B 'Eiiel yttQ ieriv cig 6 ^ ffpog %ov B, ovzmg 6 V 
srpog tav ^, o apct ftcpos ^otii/ 6 ^ tov B ^ ^c'p>7i 
To kiJto (iBQOs iarl xal 6 F rou ^ ^ t« avra ftEpl- 
^vaiAal «pa, o ftt^pog ^htIv 6 A tov T ^ (t^ipV^ ^° 
a-inb fiigog ietl xal 6 B tov ^ ij tu avru fti^i}. ieta 
10 apK wg 6 A jtQog rbv P, ovriog 6 B npog tbv ^ 
ojtBp Idsi Sst^ai. 

id'. 

'Eav mOtv ojtoeoiovv apt^&ftol xal aXlM 

avTotg foot to nA^^og evvdvo lafi^avojievM 

15 xal iv T<a avta loyip, xal rfi' teov iv rto Bvn^ 

loy^ Seovtai. 

"EeTsteav baoeowvv dffi&iiol ol A, B, r wl alka 
avTotg fffot To mA^#os evvSvo Xafi^av6(iEvoi iv xa 
avTm koya oC /1, E, Z, wg jifv o A jrpog tov 
20 ovxtag 6 ^ Tipbg tov E, tos Si 6 B jtQog tov V, ov- ' 
tms 6 E «Qog tbv Z' kiym, oti xal SC teov iOvWm 
mg b A Jrpoff toi' F, ovrcoq 6 ^ JtQog rbv Z. 

'Entl yaff ieTiv mg b A ffpo? tov B, ovtas o \ 
TtQog zbv E, ivalKai, a(/a ietlv as b A npog tow J 
2B ouTms 6 B jrpog Tbv £^. naAtf , ijttC ietiv ag b% 
ffpog rov Fj ovTojs 6 £' Jrpos rov Z, ^vaAAa| i 
^ijTiv ag B itQog tbv E, ovTtas b F Jtpog tbv j 
d>s S\ B nQog rov S, ovTcog 6 A TtQos t^bv /d' 



S. B] e oorr. V. fi^pij td a-uio: p. 15. nai] om. Va 
Idyoi] m. rec. B. 17. r] r, J p. 27. tos] om. p. 



B 



ELEMENTORUM LIBER VH. 219 

Sint qnattuor nnmeri proportionales ^, B, F, J, 

ita ut sit ^ : B BB F : ^. dicO; esse etiam permutatim 

^ : r = S : z/. 

^ nam quoniam est A : B — F : ^, quae 

pars uel partes est A numeri B, eadem pars 
uel partes erit etiam F numeri ^ [def. 20]. 
itaque permutatim quae pars uel partes 
est ji numeri F, eadem pars uel partes 
est etiam B numeri ^ [prop. X]. ergo 
j^ : r = B : ^ [def. 20]; quod erat de- 

monstrandum. 

XIV. 

Si quotlibet numeri dati sunt et alii iis numero 
aequales bini simul eoniuncti et in eadem proportione^ 
etiam ex aequo in eadem proportione erunt. 

Sint quotlibet numeri ^^ B, F et alii iis numero 
aequales bini simul coniuncti in eadem proportione 



B E 



z/, £, Z, ita ut sit ^ : S = z/ : £ et S : r «= £ : Z. 
dico, esse etiam ex aequo -^ : F = z/ : Z. 

nam quoniam est A: B = /i : E, permutatim erit 
A: jdi = B : E [prop. XIII]. rursus quoniam est 

B:r=E:Z, 
permutatim erit S : £ = T : Z [id.]. sed B:E^A:J. 



220 STOIXEIiiN £'. 

wg npK 6 A wpog zov ^, otTos 6 F Jtffog tov 
ivalla^ aQtt iaxlv mg 6 A Kpog -"" ^ 
nffbg tbv Z' ontp i'iJ« d'£r|«t. 




1/ r, ovTms 6 



5 '£(iv fiovtts aQt&ftov tiva [liTe^, teaxig 8i 
iTipos i(gi&(ios «cAAov Tivu aQt&jiov (lErpy, xal 
ivaXXa^ iodxis ij (lovas loj' TptTOf detO^jiov 
^ETpijHEi xal 6 davTe^os tov r^repTOv. 

Movag ycci) 57 ^ ap(#{[tdv ztva tov BJ' fietQeiTa, 

10 iacixis Sh izegos aQi&ftog ^ aAAor Tii/a d^i&fMiv 
Toi- -EZ fiST^titm' ksyBi, ot( xal ^vcAAkI (adxis ^ A 
fiovKg tbv ^ aQt&fiov fiiTifsl xcl o BT zbv EZ. 

'ETiel ywp iedxis n A ftovas tov BF aQi&fiOV 
fi£Tp£t xal i jd Tov £Z, off«(. «p« «^olv iv Tp Br 

15 fiovdSig, ToOovTot E^ffi «cl iv za E2, dQi.&fi.ol tSot 
T(5 A. dtijpijtfd'» 6 ftiv BF ilg xkg iv iavr^ fto- 
vdSas TRs BH, H&, &r, 6 Si EZ els zovg za A 
taovs zovs EK, KA, AZ.. ^azat Sri taov t6 TckijQos 
tSv BH, H&, @r zm al7}&£i tav EK, KA, AZ. 

20 x«l ixsl raat sieXv at BH, H@, @r fiovdSes «^Aij- 
lats, eial dh xal of EK, KA, AZ aQt&fiol iaoi dXX^- 
lots, naC iativ i'Bov tb nlij&og tav BH, H@, @F fio- 
vdSav tm rrAtjftEt zav EK, KA, AZ aQt&ftav, iUTiw 
apa ©s ij BH jtovds jrpog tov EK aQt%^fi6v, ovxas 

26 ^ H® fiovdg Jrpog Tov KA KQtQfibv xal vj &r ^o- 
vig ic(>6s zov AZ aQt&ftov. iatat ctQcc xal cos 



Z.huildl ceo] in ras. : 
upi&fidv] om. p. 7. Ufi9n6i 



■ » p- 



n 



8. /lE-ze^i B. 9. tlta] 
10. di] supra m. 1 V, _ o J] 



■ KLEMESTOKOI LIBER VII. 221 

-quare etiam j4 i ^ '^ V : Z. ergo permutatiia erit 
j1 : r= ^ -.Z [id.] ; quod erat demonetrandum. 

5V. 

Si nnitas numerum aliquem metitur, et alius nu- 
merus alium numerum aequaliter metitur, etiam per- 
mutatim unitas tertium numerum et aecundus quartum 
aequaliter metietur. 

Nam unitas ji numerum aliquem itrmetiatur, et 

alius numerus ^ alium numerum EZ aequaliter me- 

tiatur, dico, etiam per- 

, ^ , f , ^ f mutatim unitatem jS nu- 

^ merum ^ et BT nume- 

, _ rom EZ aequalitei- me- 

I- — I 1 1 tiri. 

nam quoniam unitas ji numerum BV ti tJ uume- 
rum EZ aequaliter metitur, quot sunt in BT unitates, 
tot etiam in EZ numeri sunt numero ^ aequales. 
diuidatur BTin unitates suas BH, H®, ©Tet EZ in 
numeros EK, KA, AZ numero ^ aequales. erit igi- 
tur multitudo numerorum BH, H®, @r multitudini 
uumerorum EK, KA, AZ aequalis. et quoniam 

BH = H& = ®r 
et etiam EK = KA = AZ, et multitudo unitatum 
BH, H®, &r multitudini numerorum EK, KA, AZ 
aequalis est, eiii BH : EK = H& : KA = ®r: AZ. 

IttTf/thio] om. V f. fottxis] om. p. 12. fitrQei 

16. eloiv PB. afi^tia p. 16. d] tj qi, tuv- 
ttvxu Vpq>. 18. Sj}] 6b p. 10. KA] K e corr. V. 
' EK] zco M. EK f. 24. ros] m. 2 V. triv] 

o5ii»e] 'in ras. ni. 2 V. 26. ff 6] in rag, m. 2 V. 
M.A] in ras. m. 2 V, xol ^ — 26: uti^fiov] mg. m. 2 ¥. 
- 26. dti&y4v\ om. B. iaxai] Sariv conp. p. 



222 iTOKEiiiN E", 

Ttav ijyoviidvav npos tva rav tnoiiivtov, outioi,' tixav- 
T*s of ^yov^ivot nffog Sxavttts rovg inofidvovs' «jiiv 
agtt ag ij BH fiovas xqos tbv EK RQt&fi^v, ot^os 
6 BF srpos tbv EZ. terj Si ij BH (lovces t^ A (lo- 
6 vadi, 6 di EK aQL&^ibs tp ^ agi&ii^. iertv aga wg 
^ ji [lovas Ttffbs tbv ^i afft&iiov, oSrats o BF itpis 
rbv EZ. laaxis apa 17 A fiovas zbv d afft^itov ( 
TQit xal 6 Br EOt/ EZ' offtp IStt dtt^ai. 



10 Eav SvQ affi.&^ol %oHanXa6idaavr€s 
kovg xoiaaC rivttq, ol ysvofiEvot i^ avzmv ftfl 
«iAijAotg laovrai. 

"EarioiSav Svo uQtffiiol oC A, B, xal 6 fihf ji 
B xoXiajii.aeittaas rbv F noiECrto, 6 6i B rbv A Jto. 

16 Ttkadiueas xhv ^ noistTa' idyio, oxt fffos dazlv 6 
ra ^. 

'ExeI yaQ A rbv B xoXlaiciagiaaas 
itoC^xtv, 6 B uQtt rbv T nsTQtt xark rag iv xa A 
(lovttSas. (iirfftt St xal ^ E fiovas tbv A afii9iuiv 

20 «ttru rag iv avr^ [lovadas' (adxis Squ ij E fto-i 
rhv A d(fiQ(ibv fttTQst xal B rhv F. ivaMa^ 
ledxis ij E [lovds rbv B UQiQ-ftbv iterpii xal 
foi' r. zdliv, izfl 6 B rav A TioXXttTilaaideag rbv 
jd Tit%oCy[xtv , 6 A fiQtt tbv A [ttzQsi: xara rcig iv 

25 T^ B [lovttSag. (isrQtC 6i xal i, E (Lovds tbv S 
xatd tag ii' aurp fiovdSttS' iadxig ^QCC i) E [1.0 
rbv B dffid^fibv [itTQtt xttl b A rbv /1. iedxis 
1) E fiovds Tov B d(ft&(ibv ifiitQBt xal 

3. £fa] Sfa %«i p. npo^ bia P. 4. 6] fi p. 
va.Si\ -Si iu r»s. V. 7. ij] P. A] aapra m. 2 Y. 



7(iOV 

ovi^M 

m 




ELEMEKTORUM LIBEK TO. 223 

erit autem etiam, ut uiius praecedeutium ad unum se- 
quentium, ita omneB praecedentes ad omuee aequenteG 
[prop.XII]. qoare BH:EK=Br: EZ. sed BH= A, 
et EK = j3. quare erit A: jJ = BF: EZ, ergo uni- 
tas A numerum ^ et BT numerum EZ aequaliter 
metitur; quod erat demonstrandum. 

I XVI. 

Si duo numeri alter altenim multiplicana numeros 
aliquoB efficiunt, numeri effecti inter ae aequalea erunt. 
Sint duo numeri A, B, et ait 

AxB = r, BxA = J. 
dico, esss T = /3. 

i -I A nam quoniam AxB = r, B 

I iB numerum T secundum unitates 

n— — -I numeri A metitur. uerumetiam 

z/i- — \ unitaa B uumerum A secundum 

I — I E unitates eius metitur. itaque 

unitas E numerum A Gt B numerum T aequaliter 
metitur. itaque permutatim unitas E numerum B et -4 
numerum f aequaliter metitur [prop. SV]. rursus quo- 
niam BX A = ^, A numerum /i secTindum unitates 
numeri B metitur. uerum etiam unitas E nnmerum 
B secundum unitates eius metitur. itaque uuitas E 
numerum B et v4 numerum A aequaliter metitur. 
uerum unitas E numerum B %\, A numerum T aequa- 

vcig] om. P. KO(&f(dy] om. P. pd(V f- H- nOKooiv B. 
11. not^Toi V, sed corr. 19. i;] sapra m. 1 p. E] e corr. p. 
20. «ix^ p. uotf] in las. Y. 21. lomis £(« F m. 1, corr. 
m, rec. 22, lea%ii\ om. p. fiowas /a«»"S p. B3. A^ in 
HW. m. 1 B. 25. im] auti P, corr. m. rec. 27. tov 

^ — 28: Nnl o A'^ om. p. 28. ^fiei^fi] P; ^it^ti BYip. 



i 



224 STOIXEI^N g'. 

icdxig &Qa i A ixdzEQOv xmv F^ jdl ^ez(fBt. E6og 
&(fa icxlv b r x^ jdi' Zxbq iSei SBt^a^. 

^Eav aQL^fiog dvo ccQLd^fiovg xolkaxkaifidcag 
6 jtOLTJ xLvag^ ot ysvofievoL i^ avxmv rov av- 
tov €^ov6l koyov xotg xokkankaCiaCd^BtiStv, 

'AQt^iwg y&Q 6 A dvo aQLd^^iovg xovg S, JT nokka- 
nka6ia6ag tovg ^, E noiBltao' kiyci^ ot^ i6tlv mg 
6 B JtQog tov r, ovtcog 6 ^ TtQog tov E. 

10 ^Eicil yccQ 6 A tbv B nokkanka6ia6ag tbv jd tcb- 
jtoCrjxBv^ 6 B &Qa tbv ^ (iBtQBt xata tccg iv tp A 
(lOvaSag. (iBtQBt Sh xal ^ Z (lovag tbv A aQi^iibv 
xata tag iv airtm ^ovdSag' l^dxig &Qa ri Z [lovag 
tbv A aQLd^^ibv fiBtQBt xal 6 B tbv ^, S6tiv &Qa 

Xh &g ri Z (lovag utQbg tbv A aQi^^6vy ovtwg 6 B 
^Qbg tbv /d. Sid td avtd Sii xal Sg '^ Z iwvdg 
^Qbg tbv A aQLd^iioVj ovtcog 6 F utQbg tbv E' xal 
d)g &Qa 6 B JtQbg tbv ^, ovtcDg 6 F ^Qog tbv E. 
ivakkdi, &Qa i6tlv mg b B TtQbg tbv F^ ovtog 6 A 

20 '^^bg tbv E' oTtBQ ISbl SBt^ai. 

iri . 

^Edv Svo dQid^fiol dQcd^fiov tcva nokkanka- 
6id6avtBg 7tocc56£ tivag^ oC yBvofiBvoi i^ av- 
tmv tbv avtbv si,ov6i koyov totg Ttokkanka- 
26 6vd6a6cv. 



1.6^] om. p. tav'] zov p. 6. -xov avxQV^ snpra V. 
7. noXXanXaaiaa&stci p. 8. tovg'] in ras. V. 11. t©] avvm P, 
avTco xm m. rec. 13. avxri p. 16. 17] supra m. 1 p. dqi&' 
fiovj om. P. 17. xal mg — 18: nqoq zov JE] om. P. 18. 



ELEMENTORUM LIBER VH. 225 

liter metiebatar [p. 222 , 22]. itaque A utrumque 
numerum T, J aequaliter metitur. ergo r—/t'^ quod 
erat demonstrandum. 

xm 

Si numerus duos numeros multiplicans numeros 
aliquos efficit^ numeri ex iis effecti eandem rationem 
habebimt; quam habent numeri multiplicati. 

Nam numerus A duos' numeros B^ F multiplicans 
numeros /tj E efficiat. dico, esse B: r= ^ : E, 

1 \A 

B I 1 r 1-^ 1 



-I I 



quoniam enim A numerum B multiplicans z/ e£Pe- 
city B numerum jd metitur secundum unitates numeri A. 
uerum etiam Z unitas numerum A secundum unitates 
eius metitur. itaque unitas Z numerum A et B nu- 
merum A aequaliter metitur. quare Z : A = B : ^ 
[def. 20]. eadem de causa erit etiam Z: A = F: E. 
quare etiam B : A = F : E. itaque permutando [prop. 
Xni] B : r = A : E] quod erat demonstrandum. 

xvm. 

Si duo numeri numerum aliquem multiplicantes 
numeros aliquos efficiunt, numeri inde effecti eandem 
rationem habebunt; quam multiplicantes. 

tov J^ J Y q>. 24. ixovai P. nolXanXaaidattai p, noXXa- 

nXaawiovai Y 9. Dein seq. in V : dvo yag aQi&iiol ot A^ B 
dffid^fLOv Tiva zov r noXXanXaaidaavtsg noimac tivag ot ysvo- 

Ssvoi l£ avTcoy tov avtov ^^ovai zoig noXXanXaaiaaa (ras. 2 
tt.); punctis del. m. 1. 

Euolidei, eddL Heiberg et Menge. U. 16 



226 rroKElflN f. 

z/uo ya(f Kpt9)iol oC A, R agi&ftov rtva tov F 

noUaxXaaiaeavzes ^ovg J, E icouCtaaav' idyta, 3a 

ierlv mg o A wpog rov B, ottmg o ^ n^bg tov S. 

'Ejtsl yap o A zbi' P jcokXaitXaeiaaas tov id lU- 

h aoi^MV, xttl 6 r aifa xbv A nolXaTiXaauiaag xbv J 

jcijtoiyxiv. Hia ta avtu Sij xal o r tbv S noXXtt- 

xkaaiaattq rbv E nenoiiixiv. ilpt&fibg Sij 6 F dvo 

aQt9(tovs zovs A, B noXkaitXtteiaaas tovg -d^ E xi- 

noit]xev. laziv aga mg 6 A irpog tbv B, ovtag o 

10 iJ n(>6s tbv E- o7tt(/ Sdii 6eiiai. 

'Eav tiaaaQss apt&fiol avakoyov coOiv^ o 
ix nQcaTov xal tstdQtov ysvoftfvos apc^^o; 
laog latai t^ ix SEvzipov xal T^itov ysvo- 

16 fiivoy ttpiS^ita' xal iav 6 sx «(tdrot) xal T6irap- 
TOi» ysvofisvos aQi&(ibs iaog ?; tm sx devzii/ov 
xal T^trov, of tiaaaifss aQi&itol avaKoyov 
iaovtai. 

"Eettoeav tiaaaQfg aQi&itol dvdioyov ot A, B, P, 

20 z/, ms b A Tigbg tbv B, ovtcas o F npo$ tbv ^^. 
xal b jtiv A tbv A noXkaakttaidaas tbv E aoiEltmfm 
Sl B tbv r aoXXanXaeidaag tbv Z «oititia- Xiyt 
oti. l'aos iatlv b E Ta Z. 

'O yecQ A zbv V noKXaaXaaidoas tbv H jtoieCt 

*6 iasl oiv 6 A tbv F noXXanXaaidaag tbv H neaoirixeVf * 
rov Si ^ jcoXXaaXaetaatts tbv E xsnoiTjxsv, aQtQ- 
(tbs 8ii b A Svo dgiQ^itovg zovg F, ^ aoXXaxXaetdeas 



1, 10» r] om. p. 2, rov r lOiie p. moiiJTWiiatv cp. 

as inTiv p. 6. nai} m, 2 V; om. p. Sea] del. V, oe 

6. JiM ra — 8: jUTCoiTpdv] ing> ra. 2 Y, om. ip, 7. 



ELEMENTORUM LIBER Vn. 



227 



Duo enim numeri Ay B numerum aliquem F mul- 
tiplicantes ^, E efficiant. dico^ esse A: B '^ J : E. 

I — I ^, , nam quoniam A nu- 

2 , , merum F multiplicans nu- 

j , , merum ^ effecit, etiam F 

, , numerum ^ multiplicans 

numerum ^ effecit [prop. 
XVI]. eadem de causa etiam F numerum B multi- 
plicans numerum E effecit. itaque numerus F duos 
numeros A, B multiplicans numeros ^, E effecit. 
itaque erit A : B = A : E [prop. XVII]; quod erat 
demonstrandum. 

XIX. 

Si quattuor numeri proportionales sunt^ numerus 
ex primo quartoque effectus aequalis erit numero ex 
secundo tertioque effecto; et si numerus ex primo 
quartoque effectus aequalis est numero ex secundo 

tertioque effecto^ quattuor numeri 
proportionales erunt. 
3 Sint quattuor numeri propor- 

j5 Z tionales A, B, Fy ^, ita ut sit 

A : B = r : A, et AX^ = E, 
Lji Bxr= Z. dico, esse E = Z. 

nam sit ^ X r= if. iam quo- 
niam AxF = H et AxA — E, 
numerus A duos numeros F^ ^ mul- 



HL 



euan. V. 11. t^'] om. qp, nt semper deinceps. 18. n^t^- 
^S^ om. p. 14. Ix dsvteQOv] PB, i% tov SevtsQov Yq); 

dsvtiifO) p. tQizo) avyHsifiiva dQi^ii^ p. 17. aQid'(ioi'\ 

om. p. dvdXoyoi p. 21. £] in ras. Y. Post noieCzto ras. 

3 litt. V. 26. nenoiTiTis Ytp. 26. 9i] snpra Y. 

15* 



228 



STOIXEiaN J'. 




Touff H, E nsTioi^xtv. sativ a^a (og o r" lipAg xov 
/i, ovTms M XQOS rov E. «Ai' fog o T Jtp6g riv 
^, O0TWS » -^ «pos Toi' B- xal mg &Qa 6 A Jtpos 
Tov Bf ovrag o H npog roi' E. aakiv, insl 6 A 
6 Tov r noli.aalaaia0tts rov H iiezoitixsv, aKla (iijv 
x(d o B Toi' r izoXi.ttnXtt0iiiaag rov Z TceTtoiTjxtVf 
6vo di] dgi&fiol ol A, B ttQi9[i6v ziva rof F jcoi- 
XankaffiaaavTsg loug H, Z icsicoitjxtteiv. ieriv agtc 
log A repos rbv B, oiTag o H itpog zov Z. aXltt 

10 /lijv xal mq o A itQOg zov B, ovrois o H jtQOs tov 
E' xkI (hg apa 6 H XQog xbv E, ovrtog 6 H «poj 
Tov Z. H apK 3t(fbg ixatBQov rav E, Z tow ai 
^X^t loyov tOog aQa iatlv 6 E ra Z. 

"EffTiD dij niihv l'0og 6 £ rp Z" Isya, oti it 

15 as o A itQos tbv B, ovte>s 6 F ^rpog 

Tiov yuQ Kvrmv xttraaxsvaS&ivzatv , ixel tOog 
iatlv b E ta Z, iariv aQtt ag b H :ip6j rov E, 
ovtas o H TtQbg tbv Z, £XX' ag (i-iv 6 H JtQog 






rov E, ovros 6 F HQog rbv 

) Tov Z, ouTojg o A Jtpdg Tof 

sipos rov B, ovros o F nQog i 

> H- 



8\ H itQbs 
B. xttl tag UQa 6 A 
IV ^' oncp ISet Sel 



1. a V. 



2.H]Ap. 
A nfog zov B, 
" 4. oviiog -xai p. 



2. OBlQJg 

ms] P; m Si Bp))!. 3. »«) . . , 
oSiois *« V, om. fp. mg apa] inojiEe P. 
H wpos TOf B] om. 9. _^Post E in V aaa. m. a: otg o a 
wpos TOf B. Hic if> mg,: oiiiojE 9i khi 6 H nifig zov E as 
6 A «poe To»i B. 0. neMo6jKf Vqj. 12. eKaztfa ip. 16, 
inE^] del. m. rec. P, adacripto lc/nEi. Font Inii add. Tp^: 
o A xovg (n^og rous p) r, ^ noUvnlttfotvaas tovs .H, E xi- 
xoiij^tv, intiv afu los P jigos zov d, ovTiag H vQoe toi 
£; idem praemiBSO iitei P rag. m. rec, et item praenuBBO 
inei et additjs: iiiofi Si laziv o £ tcS Z' forw u^u os H 
npOG TOV G B mg. m. 2, deletis lin. 16: taog imiv — 17: tov E. 
l'aog] ftofi ie Vpgj. 17. ietiv] mutat. in *e m. rec. P. E] 



9 

3r«s] 



ELEMENTORUM LIBER VH. 229 

tiplicans uumeros Hy E effecit. erit igitur 

r:d = H:E [prop. XVII]. 
uerum F: ^ = A\ B. quare etiam A:B = H:E. 
rursus quoniam Axr = HeXBxr=Zy duo nu- 
meri A, B numerum aliquem F multiplicantes nume- 
ros Hj Z effecerunt itaque A:B = H:Z [prop. XVni]. 
uerum etiam A:B = H: E, quare etiam H:E = H:Z. 
H igitur ad utrumque E, Z eandem rationem habet. 
ergo ^ = Z [V, 9]. 

Sit rursus E= Z. dico, esse A: B = r: /J. 

nam iisdem comparatis quoniam E= Z^ erit 
H:E = H:Z [V, 7].^) 
uerum H:E = r:A [prop. XVII] et H:Z = A:B 
[prop. XVni]. quare etiam A: B = F: ^] quod erat 
demonstrandum. 



1) Cnm Euclides plerasque propositiones libri V propria 
demonstratione usus de nnmeriB itermn demonstranerit, in 
qnibnsdam hoc neglexit, nelnt V prop. 11 in his propositio- 
nibus saepissime utitur, p. 228, 13 einsdem libn prop. 9, 
nostro loco prop. 7, et similiter in aliis. 



e corr. m. 1 p. ^ativ aQa — 18: tov Z] mg. m. 2 V. iattv] 
iati 9. El Z 9. ,18. ^Z] E 9. 19. ^] in ras. V. 
Post h add. Yp9: %al mg aqa 6 F mfog tov z/, ovtos 6 H 
nQog tov Z; idem inser. B m. 2, mg. m. rec. P. 20. %a£'] 

om. V9. 21. Seqnitur in Vpqp propositio de tribus numeris pro- 

Sortionalibus; om. P m. 1 (in mg. adscripsit m. recens) et 
ampanus (u. p. 231 noi); in B in mg. legitnr a manu 1. 
itaque Theoni tribuenda esse uideri potest; u. appendix. 




VMM 



Ot iXaxi6toi aQii^nol tav rov a^Tov l6 
yov ixovtmv avtotg (tEtffovai tovg tov avtov 
loyov i'xovtas iadxi^ o re p,Ei^&v tov /tfigorot 
5 ical 6 ilaeaav tbv iXaeffova. 

"Eataottv yttQ iiittxitstot dpi&jiol tav tov owroi' 
idyow ix^vtav totg A, B ol FiJ, EZ' Xiya 
laaxtg b P^ tov A iietqbi xul & EZ tov B. 
'O r^ yaQ ToiJ A ovx ioti ^iQr}. sl yiff 5i 

10 ToV, iatto' xal 6 EZ apa tou B ra avta (lipi] ietiv, 
SaeQ 6 FA tov A. oea apw iatlv iv ira PA (liqn 
tov A, toaavtd iati xal iv rra EZ liigt] roi? B. 
St.rjg^a&a b fiiv FA tlg to: roy A fiiQf] rd VH, Hjd, 
Ss EZ bIs td xov B fi^pjj t« E&, &Z- iatai Jij 

16 taov ro Til^&og tav FH, HA tm jEAij*f( tmv EB, 
&Z. xal ixBl taoi Elalv o( FH, HJ aQi&ftol al- 
A^iotg, atel S^ xal ol E&, ®Z cptd^oi fffot aXX%- 
Aoig, xal iativ taov to reA^^&og itov FH, HjJ tp 
jtX-^&Et tmv E&, @Z, iativ ttpa tog 6 FH Jtgbs tov 

20 E&, ovtats b H^ jrpog lov @Z. larai aqa xal ag 
eIs tcov iiyovfLBvav itpoff Eva tmv ESOftivav, ovras 
ttXavxBs ot ijyovnEVOi Jtgbg anavtag tovg EJlOftdvovs. 
Sativ aga tog o FH ffpog tow £©, ovttag b Fd 
«pog Tow EZ- of FH, E@ aga Tofs FA, EZ 



1. *'] %a' Vp9, P m. rec; in B non Uqnet. 8. 

A] coiT. ex 10 A V. 9. lauv B. 10. loitD 6 rj 

A (tigTi Vpq), loii Bl poatea add. V. 11. oTcig B, i 

m, 2. 1% hnv B. tov} bis V. 14. ©Z] 0H P; < 
EQ. yec. (aov Sii Saxui p. d^'] in ras, 7. 16. 

*)iet — 19: tmi' E0, BZ'] del. V, om. q). IG. taoi liat^ 
om. V. ttll^loie] foot uUiilate «wiV V, 17. ttaC~\ s" 
ftfoi p. Tiwt] om. p. 19. E®] © e corr. V. 2! 
T«s P, corr. m. rec. 



ELEMENTORUM LIBEE VII, 



231 



X.') 



H Numeri mitiimi eorum, qui eaiideni ao ipai ratio- 

* nem habent, numeros eandem ratiouem habenteB ae- 

qualiter metiuntur, maior maiorem et minor minorem. 

Sint euim Tz/, EZ minimi numeri eonim, 

qui eandem rationem habent, quam A, B, 

dico, r^ numerum A et EZ numerum B 

aequaliter metiri. 

nam F^ numeri j4 non est partes. nam 
si fieri potest, sit. quare etiam EZ numeri 
B eaedem partes sunt, quae F^ numeri A 
[prop. Xni, def, 20]. itaque quot sunt in F^ 
partes numeri A, tot etiam sunb in EZ nu- 
meri B partes. diuidatur F^ in m, H^ par- 
tes numeri A, EZ autem in E@, ©Z partes 
numeri 3. erit igitur multitudo uumerorum 
m, H/i raultitudim numerorum E®, ®2 
aequalis. et quoniamrff^if^ et £® = ®Z, 
Bt multitudo numerorum FH, H^J aaqualis est mul' 
liitudini numerorum E&, @Z, erit 

rH:E@'='H^:&Z. 

lare etiam ut unus praecedentium ad unum sequen- 

lium, ita omnes praecedentes ad omnea aequentes [prop. 

XII]. quare Fif : £© = T^ : EZ. itaque FH, E& 



1) Db propoaitione hio omiBaa haeo habet Campamus VU, 20 
add.; Qoa proponit aiitem Euclidea de tribuB numeria oontinQe 

Sroportioiialibua , quod ille qui ex ductu primi in tertium pro- 
QCitnr, Bit aequaliB qoadrato medii, et ai ille qni ex pnmo 
in tertium producitur, &erit aequalia quadrato medii, quod 
illi trea numeri aint continue proportioualSB, sicut propomt in 
16 Bezti de tribus lineis. hoo enim facile demoDatratuc per 
hanc 20 cetL 



li 



232 ETOIXEIfiN £'. 

TH «vrm i.6yp elelv iXaeuovtg ovTsg avTtSv ojTEp 
ierlv ttdvvazoV VTToxHvrai ya^ oC F^. EZ eXaxioroi 
tmv thv avtov koyov ijfivxfav aviols- ovx uQa fidfiTi 
iatlv 6 r^ rotj j4' [lifiog apa. xctl o EZ tou B t6 
6 avto jit^pog iOtCvy onsp o F^ toii A- iaaxig apK 
o V^ xhv A fietffit xal 6 EZ tov B' SnEp Mbi 
Sitkai. 



OC aQarot ffpog aAi.-^Xovg api&ftol iXaxteTt 
10 list tav Tov avTOV Xoyov ixovtmv avtots. 
"EatGitiav npmt 01 TiQog all^lovs &pt&ii.ol ot 
B' ^.iya, oti ol A., B iKa%i6tf>l eldi rmv tov av 
koyov ifflvxBiv Rvtots- 

Ei ycLQ f(^, ieovtai ttvtq rav A, B iXaatsovi 
15 apcftfto! iv rm avtip loya ovKf zots -^) B. loriaaetv 

oC r,j. 

'Easl ovv of ilaxiOTOi agi&ftol Tror tov avrof 
loyov ixovtav furpovoi tovg rbv avtov koyov Sxfiv- 
TBg iattxi.s o Tf ^d^iov roi' fi,tit,ova xal h iXattav 
20 Tov ilatrova, rovriariv o te Tjyovfitvos tbv 17; 
{ttvov xal 6 inofitvos tov imofttvov, iodxtg uQtt & 
tbv A itstQet xal b ^ rov B. oaaxis dij 6 P 
A (ttTQst, toaavrat novdSsg iatatoav iv r0 E. 



[ 



1. flfreel Oin- V- 2. igtiv} P, om. BVpip. 3. 
om, B. evTov] om. g.. 4. EZ] P; EZ Sga BVpgi. 
t«a%iS Sftt 6 rd rov Al mg. ip. Seqnitor propoaitio (,.._. 
dam nona in BVpgi, a Th«one interpolata; om. P (add. mg. 
m. rec.) et Campanae (u. p. 333 not.}; u. app. 8. Ktt'] ny' 
BVp, P m. rec. 10. elatv PB 12. eiaiv F. _ 14. Port 
(i^ add, Theon: i/oi» 0! A, B ^lajiiafai TtB»> toy «vTOf l^a)| 
^o'»!!»»' auiot"e (BVpip). 15. B]_^ corr. ei T m. 1 p. 

jilaaffafa Vp^. tdvt^cti if. 



m. 






ELEMENTORUM UBER Vll. 233 

tninoreB mimeris P^, EZ in eadem proportione simt; 
quod fieri aoii potest; naiu supposuimus , Fz/, EZ 
minimos eeae eomm, qui eandem habeant rationem. 
itaque Fz/ non eat partes numeri ji. pars igitur erit 
[prop. IV]. et EZ numeri B eadem pais est ac fz? 
numeri A [prop. XIII; def. 20]. ergo F^ numerum 
A et EZ numerum B aequaliter metitur; quod erat 
demonstrandum. 

XXI.') 

Numeri inter se primi minimi sunt eorum, qui 
eandetu rationem habent. 

Sint primi inter ae numeri j4, B. dico, 
numeros A, B minimos esse eorum, qui 
eandem rationem habeant. 

nam si minua, erunt numeri aliqui mi- 

nores nnmeria ^, B, qui in eadem pro- 

■ T portione sint ac j4, B. aint F, ^. iam 

-i- quoniam tmmeri minimi eorum, qui ean- 

dem rationem habent, eos, qui eandem ra^ 

tionem habent, aequaliter metiuntur, maior 

maiorem et minor minorem [prop. XX], h. e, prae- 

I eedens praecedent«m et sequena sequentem, J" numerum 

E44 et /i numerum B aequaliter metitur. quoties igi- 

r'inr T numerum A metitur, tot sint iu E unitates. 



1) FropOBitioQein omissae Himilem habet Camptmua iu addi- 
tamento bdo VII, 19: bic ant«in demonatrare nolumiis aequam 
proportiDnalitatem in qnotlibet numeria dnomm ordinnm in- 
oirectae propartiotiaIitB.tia , qnam demonBtrat Euclidea per S3. 
qninti in qn&ntitatibas in genere. dicimna ergo; ei qnotlibet 
numeri totidem aliis fuerint iadirecte proportionalee, estremi 
qnoque ia eadem proportione proportionalea emnt, cett. 



k 



234 



ETOIXEIiiN S'. 




b iJ uQtt rof B fitxQel xara rag iv t£ E (uvSeSg^ 
xal iatl 6 r tov A (laTQsl xaia zag iv Tp E fiovd- 
Sag, xal 6 E aga zhv A jUTgit xatk xkg iv xa T 
fiovaSag. Sik tk avTo: d^ 6 E xal tov 3 ftetgii: 
6 XKta Tceg iv zw ^ (wvadag. b E a^a rovs A, B ftt- 
rpcf atfarovs ovtaq zpog all^lovg' ontQ isrlv ddv- 
vttzov. oi>x af/a SaovxaC rives tmv A, B iXdaaoveg 
affi&fiol iv Tt5 «utro Aoyto ovtag tolg A, B. oC A, 
B aqa iXi%i.<itoC iim, twv rof «uroi' Aoyov i%6vtm— 
10 avTolg' onE^i i8t<. 8hK\ai. ^fl 



Ot ikaf_i.aToi di)i&ft,Q\ xmv xov avtbv loyov 

ixovzav avzotg itffiBTOi nj>og aXX^Xovg elaCv. 

"Eataaav iXd%taxoi d^i&(iol tmv rov avzbv X6- 

IS yov ixovtav avrois oC A, B' Xiyea, Sri oC A, B 

ZQtoxot aQog dXX^^Xovs sCaCv. 

El yap fnj siai jrprarot jrpos dXXrjXovg, fisr^i^eH 
xig avtovs d(fiQfi6g. fttziffixm, xal Sarm 6 F. xa\ 
badxig fihv 6 F xov A fisxpit, xoaavxui fiovddei 
20 ietaiaav iv xm ^, baaxig de b F zov B iiEzgst. 
aavrat ftovdSeg saxaOav iv xa E. 

'Eitel 6 r xov A fiexpEl xark rds iv n 
fiovdSag, o V aQa tov A TtoXXaitXaaidattg row 



yxii . Alexander Aphrod. in anal. pr. foL 8T. 



xal 

1 



2. vttl iTtti — fioniJuel om. P (abesge non posHnat). 
£] supni if>. 4. ttt avtd] luvtu B. & E ntU] %ai b 

E Vip. 9. tlaiv PB. 11. %3' BVp, P m, rec, 

avrmv P, corr. m. 1. 13. avtolg] om. Alexaoder. 

Post: izovtmv in ¥ eUTJloit delet. 16. clei ip. 17. «^wj 
jigiStoi] oi A, B Kgttiioi Bp. liUijlove ai A, B Yip. 




ELEMENTORUM LIBER VH. 235 

quare etiam J numerum B metitur secundum unitates 
numeri E, et quoniam T numerum A secundum uni- 
tates numeri E metitur^ etiam E numerum A metitur 
secundum unitates numeri F [prop. XV]. eadem de 
causa E etiam numerum B metitur secundum unitates 
numeri ^ [prop. XY]. itaque E numeros A^ B meti- 
tur^ qui primi sunt inter se; quod fieri non potest 
[def. 12]. itaque non erunt numeri quidam numeris 
A^ B minores^ qui in eadem proportione sint ac A, B, 
ergo A, B minimi sunt eorum, qui eandem rationem 
habent; quod erat demonstrandum. 

XXII. 

Minimi numeri eorum; qui eandem rationem habent, 
inter se primi sunt. 

Minimi numeri* eorum^ qui eandem rationem habent; 
sint A, B. dicO; A, B numeros inter se primos esse. 

A I 1 nam si primi non sunt inter se, 

B\-———\ numerus aliquis eos metietur. meti- 

j\ 1 atur et sit F. et quoties F nume- 

^ ' ' rum A metitur, tot unitates sint in A, 

quoties autem F numerum B metitur, tot unitates 
sint in E. 

quoniam enim F numerum A secundum unitates 
numeri A metitur, F numerus numerum A multipli- 
cans numerum A effecit [def. 15]. eadem de causa 



uvTOvg] rovs A^ B Theon (BVpqp). iaxvi] om. tp. 20. 

J3] in ras. m. 2 P. 22. kml ydg P, insl ovv Y m. 2, op. 

23. r] J Y in ras., 9. J] F ia ras. V, tp. Ante top 

ras. y^ un. V. 



236 



rroixEiflN s'. 




siiaoirjxtv. dut ra avra Si] xal 6 V zov EjtoXi 
TtlLaeiaeas rbv B xexottjxiv. «ptfrftog S^ 6 T dvo 
aQt&fiois tovi z/, E itoXXaxlaOtaaag tovg A, B xe- 
noCrixiv Imiv aqa mg 6 jJ npog tov E, ovrms o 
f> A :Tp(s xov &■ ol A, E apa xotg A, B iv t^ ama 
Koya eialv iXdtffovEq ovttg avrav ojrep iatlv aSv- 
vttzov. ovx «pa Tovs A, B «pt&ftovg aQi9fi6s t<s 
[leTff^an. ot A, B a$ia ngatoi. Jipbs itll^Kove eieCv 
oxBp edti- dit^ai. 



10 



xy. 



'Ea 



npog ttXX-^Xovs 
api&itos JipOff 



Svo ttpi&iiol npfdiot 
taaiv, Tov tva avrmv fiet^m 
rov XoiTtov Jtffatos ^atai. 

"Eataaav Svo agi&fiol :r{i(OToi wpts aKXi^lovg ol 
15 A, B, Tow Sh A (iiXQelrm tig api^d-ftog 6 J"' Jjyw, 
OTt xal ot r, B spraioi npog aXl^^Xovg eMv. 

El yap f(»i elaiv oC F, B Jt(/mxot npbg aXX^^Xovg, 
(tsrpijffft [Ttg] Towg r, B aQi9fi6g. ftsrps^rta, xal istca 
6 A. inel 6 ^ tbv F fiitpEi, 6 Ss F tbv A fic- 
20 tQet, xal 6 A aga thv A (lexQEt, ftEtfftZ Se xaX 
TQV B' b ^ aga tovg A, B [letgei HQmtovg ovt«s 
Mpos ttXXTilovg' onep iatlv aSvvatov. ovx aQa towg 
r, B aQL&fiovs aQi&fiog tig fietQ-^aet. oC f, B 
KQoatoi JiQog aXXjiXovg elaiv o«eq SSei SEt%tti. 



1. lt£JIO(IJlte V q 



rl mntat. 
mitSftos] mut. 
ras. V", (p. i 



1 E V; I 

tp. s. atji&^bv 

Ttataffynaiitv in rai 
m. rec. 12. xiatos «tfos t^' 

leyoj poat ras. P. 18. ue] m. 

Post B add. V: a@i#ftovf, idem 



: Kp« T 0t>C I, 



ELEMENTORUM LIBER Vn. 237 

erit etiam Fx E = B, itaque numerus F duos nu- 
meros ^, E multiplicans numeros A, B effecit. erit 
igitus ^ : E = A: B [prop. XVII]. itaque ^, E nu- 
meris A, B minores in eadem proportione sunt; quod 
fieri non potest. itaque numeros A, B nullus nume- 
rus metietur. ergo numeri A, B inter se primi sunt; 
quod erat demonstrandum. 

xxm. 

Si duo numeri inter se primi sunt^ qui alterum 
eorum metitur numerus, ad reliquum primus erit. 

Sint duo numeri inter se primi A, B, et 
numerum A metiatur numerus aliquis F. 
dicO; etiam F, B inter se primos esse. 
j nam si F, B inter se primi non sunt, 
T numerus aliquis F, B metietur. metiatur, 
i B r ^' quoniam A numerum F meti- 

tur, et r numerum A metitur, etiam A 
numerum A metitur. uerum etiam numerum B me- 
titur. A igitur numeros A, B metitur, qui primi sunt 
inter se; quod fieri non potest. itaque numeros F, 
B nullus numerus metietur. ergo F, B inter se primi 
sunt; quod erat demonstrandum. 



9 mg. m. 1. aQid^aos tovg F, B aQid^fkovg p. (iSTQr^m q>, 
19. inBH xttl inei V, inel stg q>. 81. tovg] corr. ex to 

m. 1 P, tov p. 23. r, B] (prius) B, T Vqp. 



238 



ETOISEIiiN £'. 




xS'. 
'Ettv Svo aifi&fiol JTpds tiva aQt&fXQv 3t(f\ 
xoi m6iVf xul 6 f'i avrav yevofttvog Te^dg sa| 
ttvtbv XQcoTOs itSTtti. 
6 ^vo yttQ api&fiol oC A, B ngoe ttva affi&iii\ 
Tov r 7t(fatoi ISTaaav, ttal 6 A tbv B xoJLXaaXa- 
oittaag TOf ^ xouitm' liym, ori ot P, ^ ir^m 
XQos «AAjj^ovs Bteiv. 

Ei ya^ fi»j sieiv oC F, /} hqiotoi npbs dlX^ijXovg, 

10 ftsrpjjUft [zig] rovg T, ^ d^td^iios. tiSTQiitta, xal 
iaxm E, xal ixEl ol F, A mQmrot itQos aXX^Xovg 
tioiv, Tov di r (tBtQii: Tis uQt&ftbs 6 E, o^' A, E 
ccQtt agmtot itQos aXXiiXovg eieiv. oedxis Sij 6 E rbv 
z/ fittQs!:, xo6avzt(t novaSgg SotaCav iv tm Z* xa\ 

15 6 Z aqa tov A fist(/st xarw Tccg iv Tp E fiovaSag, 
6 E Kpa tov Z noXXaitXaaideag rbv /S JlSTtoitfxsv. 
aXXa ^riv xa^ 6 A zbv B noXXaxXttaiaettg xbv A 
ns%oiri%sv ieog apa iszlv o ix zav E, Z z^ ifii 
rmv A, B. iiiv Ss 6 wjro xmv axpmv ieog y rp vsi^ 

20 tmv iiieav, ot xieeaQes aQt&^fiol avdXoyov tiaiv 
seriv Kp« as E jrpog tbv A, ovtms b B jcqos tbv 
Z. oi Ss A, E XQmxot, oC S^h Ttqaxoi xal iXaxtexoi, 
oC 6i iXdji^ietoi «ptfl^ftol tmv rov avxov Xoyov ijjoV 
zav ttvtotg ^sxQovet taiis tbv avtbv Xoyov i%ovtaq 
85 ieaxtg o xs \i.{iiav tbv [isi^ova xal 6 iXaeaav xbv 
iXdeaovtt, xovtiextv o rs ^yovfievog tbv ^yo^f 



1. «C BYp, P m. rec. 8. Fost aQi^fioi add. V ( 
et ip: n^mioi. aQtditov] mg. m. 2 T. «^mToi] om. V if 

S. aai PVp^. ttQtotoi larai npoe ziv nvroi' p. 7, «»_ 
^icD ip, sed cocr. F, d~\ e corr. m. 2 p. 10. iis] om. Fa 



i 



^M 



i^^ 




IB 



A 
Z 



I 



£ 



ELEMENTORUM LIBER Vn. 239 

XXIV. 

Si duo numeri ad numeruin aliquem primi sunt^ 
etiam numerus ez iis productus ad eundem primus erit. 

Nam duo numeri A^ £ ad numerum 
aliquem T primi sint, et sit -^ X 5 = /i, 
dico, etiam F; ^ inter se primos esse. 

nam si T^ ^ inter se primi non sunt, 
numerus aliquis numeros T^ ^ metietur. 
metiatur et sit E. et quoniam F; A 
inter se primi sunt, et numerum T nu- 
merus aliquis E metitur; numeri Ay E inter se primi 
sunt [prop. XXIII]. quoties igitur E numerum J me- 
titur, tot unitates sint in Z. quare etiam Z numerum 
^ metitur secundum unitates numeri E [prop. XV]. 
itaque E x Z = -J [def. 15]. uerum etiam AxB^d. 
itaque ExZ^ AXB. uerum ubi numerus ex ex- 
tremis productus numero ex mediis producto aequalis 
est, quattuor numeri proportionales sunt [prop. XEX]. 
itaque E:A = B:Z. sed A^ E primi sunt, primi autem 
etiam minimi sunt [prop. XXI], minimi autem numeri 
eorum, qui eandem rationem habent, numeros eandem 
rationem habentes aequaliter metiuntur, maior maio- 
rem et minor minorem [prop. XX], h. e. praecedens 
praecedentem et sequens sequentem. itaque E nume- 



add. m. rec. Post J add. Vqp: aqi/S^yi^ovq, ttQtdiLiog] corr. 

ex affi^ykov^ m. rec. P. 11. ot F, Al corr. ez 6 T 9, ex 

of r, ^ p m. 2; ot A, T P. 12. BlaC Vpy. A, E] E, A p. 
13. «9«] om. Ytp. 19. tcog] Haov (jp. 20. ot]aiC^ P, 

del. m. rec. avdXoyoi p. 26. IXaxtmv P. 26. ilattova P. 



240 



ETOtXEIiJN t'. 



xal i:i6(iivog tov mo^voV o E affu zov B fis- 
tQBt. fieti/ei di xal xov F' h E apa Tovg S, T [U- 
TQet 7IQ(6tovs ovttts J^pos aXX^lovs' ojtfp iaxlv dSv- 
Vttzov. ovx aQtt roiis F, A aQi.%^ovq ttQi&(i.6s ry 
5 (iBTQ^qeet. ol r, ^ upa JtQmzoc wpog kI).^Xovs eia(v 
OTUif ISu S£ii,ai. 

'Eav dvo ttfii.&ftol JtpioTot wpog aXXijXovs 
mtSiv, 6 ix TOv ivog avTav yevofievos JfQOS 
10 rof loiJtov jrpiMTOs Satai.. 

Barmaav Svo aQi&fioi x^atoi, nQoq akXtikovs ol 

A, B, xal o A iavzov noAAturAaffiatfag tov F Jtoiei- 

Tcj" Xdym, oti ol B, F ngaToi. tcqos «AA^Aovg eieCv. 

KeCe&a yuQ Ta A iaog 6 ^. tJtel of A^ B nqa- 

IG ro( JtQos ttXhqXovs eCtfCv, tSog Si o A ta ^d, xal ot 

^, B KQtt Jtptorot ffpog «AAijAous ^laCv. exmsQog Sqb 

tav ^, A wpos rov B jcptoTog iatiV xal o ix tav 

A, A a^a yevi.fisvog Jipos rov B jrpaiTos larai. h 

di ix tmv J, A yevofievos ttQi9fi6s iativ 6 P, of 

20 r, B aga xQmtot wpog (JAAijAous elaCv oasQ i9a 

dstiat. jH 



'Eav 
iporsQot 

5 i| ttVti 



dvo uQid-fiol npos dvo 
npog ixaTfpof n^ptorot 



Igid^fiovg a^- 
coaiv, xal oC 
og aAAijAovs 



Avo yaQ aQi&y.ol oC A, B ^Qog Svo aQt&fiovs 
Toiis r, A dfipoTSQOi npos sxareQov aQ&tot iotm- 




- B 



ELEMENTORUM LIBER VH. 241 

rum B metitar. uerum etiam numerum F metitur. 
itaque E numeros B, F metitur, qui inter se primi 
sont; quod fieri non potesi itaque numeros F, /1 
nuUus numerus metitur. ergo T, ^ inter se primi 
sunt; quod erat demonstrandum. 

XXV. 

Si duo numeri inter se primi sunt^ numerus ex 
altero eorum productus ad reliquum primus erit. 

Sint duo numeri inter se primi A, B, et sit 
j A^ = r. dico, numeros B, F inter se pri- 

mos esse. 

ponatur enim A = A. quoniam A, B inter 
^ Jt se primi sunt, et -^ = ^, etiam A, B inter se 
primi sunt. itaque uterque A, A 2iAB primus 
est. quare etiam A X A ad B primus erit [prop. 
XXIV], uerum AxA = r. ergo F, B inter se primi 
sunt; quot erat demonstrandum. 

XXVI. 

Si duo numeri ad duos numeros singuli ad sin- 
gulos primi sunt, etiam numeri ex iis producti inter 
se primi erunt. 

Nam dub numeri A, B ad duos numeros F, A 



ex AE avtov B. ^13. B, T] T, B P. f^cr^ Vpy. 14. 

%a\ ins^ V9; insl ovv p. 16. taog Si — 16: dXXi^Xovg slaiv] 
om. B, mg. m. 2 V. 16. B] in ras. Vp. noog'] an a^ 9. 

17. iaxiv^ PB; comp. p; ^cjTt V9. l8. apaj supra V, 

Ixi 9. yiv6|Lievo5 B. Post hoc iierbum ras. dimid. lin. V. 

22. yL7\ BVp, P m. rec. 23. a^i^\i.ovg\ om. p. 24. (D<Ft 
PVp9. 

Euclides, edd. Heiberg et Menge. U. 16 



242 



STOESEIiiN i 



6av, xal o ftlv A zov B TtoilttakaaiKaas xov E 

RomVo, b S^ r xov ^ aoXXanla6i.a9ttg tov Z noi- 

eCta' Xtya, ozi oC E, Z jrpcHTOt Jlpog alX^^qkovg eleCv, 

'ExbL yaff ixKTsgos tmv A, R x^s tov V xpa- 

5 rog iariv, xal 6 ix rmv A, B uqk YtvofiEvog x^iq 

101' r n^aro; itsrat. 6 Si ix %mv A, B ysvoficvos 

iotiv 6 E' ot E, r aga jiQatoi Jrpoff aXl^^Xovq eUsiv, 

Sia za avza Srj xal ot E, ^ Jtpratot irpog (cAAijAovg 

kIoCv. ixaxEfjog aqa xmv r", ^ it^os xov E stgioTog 

10 ioTiv. xal o ix xmv F, /t aqa yivo^fvog ngog zov 

E nQmtog icxat. 6 61 ix xmv F, jJ yEvofiEvog imtv 

6 Z. 01 E, Z apa n:pc5rot srpog alXriXovs elei 

07tc(f iSn tfet^Ki. 



5 'Eav Svo dgi&fiol XQmxoi sipos aXXi^X\ 
maiv, xal JioXlaTcXaeidoag ixdrtQog iavri 
xoiy rtva, oC yavofievot i^ avxwv TfQcoxot JtQos 
ttXX-^lovs leovxat, xav ot i| dpxVS Tovg ysvo- 
fi.ivovs TCoXXajtlaaideavtis JtotaeC xivag, xd- 
xsivot itQmtot n;^o? dXXijXovs saovxat [^xul ast 
nsQl tovg axQOvg Touro avfipaCvet]. 

"Eetmaav Svo aQt^ftol nptaToi npbg dXl^^Xovs 
SSVII. Alexand. Aphrod. in Einal. pr. fol 87, 

1. E — 2: jioUtudtiatdaas 10»] mg. m. 2 P. . _ 

codd. i] om. 01. •/cvaitcvog aga Vqi. 1. o E initv j 

M Ytp. 8. ditt ra — 9: claiv] TcdUv B. 8. la 

rniliK Yq>. E, iJ] i/, E P. 9. Sga] om. B. 

x6\ ip, 10. iari BV51; comp, p. 11. foiail Imi ooj 
r, d] ^, r V9. 6 Z iaxiv p. 14. «&' BVp, P n 

16. (oirt Pp. 17. tt^imii] ■mv m ras. 9). 81. ' 

corr. ex co avid m. S B. S2. Hd] supra m. S V, c^c&fU 
Jvo P. 



1 



ELEMENTORUM LIBER VH. 243 



singuli ad singulos primi sint, et sit 

AxB = E, r X ^ = z. 

dicO; etiam E, Z inter se primos esse. 



A I 1 r I 1 

B 1 1 J I 1 

E\ 1 

Zi 1 

nam quoniam uterque A, B a,d F primus est^ etiam 
^ X J5 ad r primus erit [prop. XXIV]. sed ^ X 5 = jB. 
quare Ey F inter se primi sunt. eadem de causa 
etiam E, ^ inter se primi sunt. itaque uterque Tj j^ 
ad E primus est. itaque etiam r X ^ ad ^ primus 
erit. sed Fx j^ = Z, ergo Ey Z inter se primi 
sunt; quod erat demonstrandum. 

xxvn. 

Si duo numeri inter se primi sunt^ et uterque se 
ipse multiplicans numerum aliquem effecerit; numeri 
ex iis effecti inter se primi erunt; et si numeri ab 
initio sumpti numeros productos multiplicantes numeros 
aliquos effecerint, ii quoque inter se primi erunt [et 
semper in extremis^) hoc accidit]. 



1) axpot hoc loco satis insolenter positum est. neque 
enim alind significat nisi: nltimos, nltuno loco productos. 
praeterea mirum est, haec nerba in demonstratione ne nerbo 
quidem resjpici, nednm demonstrentur. quare puto, uerba xal 
ael TTf^l Tovff ofxQovs tovto Gv(ipa£v£i lin. 20—21 ante Theonem 
interpolata esse; omisit Campanus YII, 28 (sed in demonstra- 
tione addit: sicque si infinities duceretur ntmmqne producto- 
rum in suum principium, essent omnes producti contra se 
primi; et non solum, sed quilibet eductus ab a ad quemlibet 
eductum a b). 

16* 



l 



244 STOIXEISiN £'. 

A, B, xal 6 A iavTov filv xoJLlcazlaauiaag 
jtoiiir©, rbv Si F nuXKaTiktteiaaaq tov 

da B iavzov [iiv 7CoXXaai.a0La<}as z6v E sioiEitm. 
thv 81 E noXXanXatfuiaas tov Z xoitiTO' kdyca^ on 

5 ot Tf J', E xttX ot ,d, Z npiarot srpog akK^Xovs deiv. 

'Eail yag of A, B npiBtoi wpog alXtjXovg eiuiv. 

xal o A tavibv xo\kttnXttaitt6aq zov V Jtfjtoirjxtv, 

of r, B Kprt jTQmzoi Zfios alXijXovg ildCv. ijtEl ovv 

01 r, B TiQazoi Jtpog aAAijAovs tieCv, nal q S iavrov 
10 aoXXaTiXaatttaas tov E aeaoC-rjxBv, ot T, E Spa npm- 

rot 3rpo5 a/iijAowg EieCv. ndXiv, ijctl ot A, B Kpa- 
Tot irpos «AAiJiowe siaCv, xal 6 B taurow aoXXttnXa- 
atttSttg xov E stSKo/jjxev , oi ^, E aga jrpfOTO!. JipOs" 
«AiijAows Eieiv. i:iEl ovv dvo ttpi&[iol ol A, P jrpo? 
15 dvo dgt.9fi.ovg rovg B, E d(i<p6v£ifoi wpos ixdTEQiyv 
TtQiatoi siffiv , Xttl o ix tav A, V apa ytvofiEvog ^ipos 
TOf ix tav B, E apatog ietiv. xaC ietiv o (ilv ix 
rSv Af r 6 A, 6 di ix twv B, E o Z. ot .d, Z 
KQtt ngiaroi jrpog aXX^Xovg eiaCv Sjcsq Sdtt SeS^ai. 

20 Xt}'. 

'Eav Svo api&(iol nQatoi npos dXk-^Xovg 

aaiv, xaX avva[i<p6xEpog Jipos ixdtsQov avtav 

TCQcozog iatar xal idv awttiKpotsQos itpos eva 

tivtt avttav «QiBtos j;, xal ot ii «pz^S dpi&fiol 

25 nperot wpog dXlijlovs iaovtai. 

2^vyXEta&Qiaav ydg Svo «Qi&fiol ^rpiarot srpog oiXX^J 
lovg ot AB, BF' Xiyia, oti xal awa^rpatEQog 6 Atr 
Kpos ^Jcaifpov tiBv AB, BF rcptorog iariv. 



1. ficv] om. Yqp. 2. naiiCiai] aoifi p. itoielzm xo«r ^ 
Vqi (noijj™, aed corr,, cp). 3, jkV] in raa. P. 



ELEMENT0RU4I UBER VII. 



245 



A 

4 ^I 



Sint duo numeri inter se primi A, R et sit 

^x-^ = ret jxr= ^, 

BxB^E etBxE=Z. 
dico, et r', £ et ^, Z inter se primos esse. 

nam quoaiain A, B inter ae primi 
aunt, et Ax ji^V, erunt F, B inter 
86 primi [prop. XXV]. iam quoniam 
r, B inter se primi sunt, et 

BxB = E, erunt F, B 
intei se primi [id.]. rursus quoniam 
A, B inter se primi sunt, et 
S X B = E, erunt A, E inter se primi [id.], iam 
quoniam duo numeri A, F ad duos numeros B, E 
Bingnli ad singulos primi sunt, etiam AxF ad BxE 
primus est [prop. XXVI]. et ^xr=^, BxE = Z. 
ergo ^, Z inter se primi suit; qnod erat demon- 
strandum. 

XSVIII. 
Si duo nnmeri inter ge primi suot, etiam uterque 
simul ad utrumuia primus erit. et si uterque simul 
ad alterutrum primus est, etiam nnmeri ab initio 
Bumpti inter se primi erunt. 

Componantur enim duo numeri inter se primi AB, 
BF. dico, etiam AP ad utrumnis AB, BF primum 



om. Vqj. tlei Vrp. 6. iati — claiv] mg. m. 1 V. tlai 
BVpv. 8. tloi Vpgi. ijccl ovv ~ 9: E^d^v] om. p, mg. m. 
1 V. 9. etai BVp^. 11. ttai V ip. 12. eCisi PVpg>. 

H. Imi] xnl i-nei B. 16. B\ con-. es A V. 16. ilm 

Vpip. 17. rd»i] tmv g>. ^ ieri Yip, comp. p. 20. 1' 

BVpq>, P m. rec. 32. mai PVptp. avvanifOTegov avtmv 

K(DC lx<>TE(Ov Yip. 36, awiuie9aea>' ip. 2S. iiov] t6* P. 



t 



246 ZT0IXEI2N E'. 

^otpg, fittQ^ect ris loi-s FA. AB ipt&nos. jiifrp«Vo, 
xal ^ffto) 6 ^. dxel ovv o A tovs FA, AB (lETfftt, 
xul Xoiaov aqa zov BF (isTp^set. fiCTQBt di xal rov 
6 BA' 6 ^ affa Tovg AB, BT ^STpsi xginTovg ovtbs 
xgog aXlrXovg' ojtfp isrlv dSvvarov. ovx apa toiis 
FA, AB aQt&fjbOvs agi&{i6s tig ftfTp^iTEi' ot VA^ AB 
a(fu nfftoTot npog dXl^Xovg elaCv. did ra avra di) 
xal of AF, FB agaTot jtpog dXltjXovg dcCv. 6 FA 

10 aQtt Kpog ixdrsQov ttov AB, BF rcpwrtSg iariv. 

"EOTaSav dij ndXiv ol FA, AB itQaxot itQos 
a/A^Aoiig" Xiyo, ott xal ot AB, BF n^otoi npos 
dXX^Xovg aiaiv. 

El ydg fi^ aiaiv of AB, BP jtpciTOt npog aJlAij- 

16 Aovs, fittQlleu itg Toirg AB, Br dpt&(t6g. (itTQiiiio, 
xal ierco 6 ^. xal iTCBi b .d EndrtQov Ttov AB, BV 
(LtTQtt, xal oXov Kp« Tov VA ^erpTjffft. [iSTQtt di 
xal Tov AB' o ^ apa roig FA, AB [ittQtt TrpoJ- 
Tovg ovTHs irpoff dXXi^Xovg' ont^ iatlv dSvvatov. ovx 

20 «pa T0W3 AB, BV dpi&[iovs apc&ftJs Tig fiez^^m 
oi AB, Br aga jccftotot Jipog dXX-^Xovg tioCv 
idti Stt^at. 

X»'. 

"Anas jrpwrog apt^ftos apos axavTa 
25 ^o'f, ov (i^ (lETQti:, scQmTog ietiv. 



1. r^] jr p. 2, Fj*] A a corr. p. ^B] AB mpi 
fiovf Vip. api^^dc] mntat. in agiffitavs p. 6. J] lu raa, ^. 
8. iicL ta — 9: tlvCv} mg. Y. 8. dut] Beq, raa. 2 litt. ip. 
9. on af V, ip. AF, TB] in raa. p; FA, TB Vqi. r.,*] 
.^r Vp93. 10. li»'] e corr. P. _12. kk/] Bupra T. _ .,4B] 
e corr, p m. 1. 16. BT] Sr api&fiore V^, fiEr^iJTOi 91, 









f 



ELEMENTOKUM LIBER vn. 247 

nauL si Fj^, AB inter ae primi nou aunt, numerus 

aliquia numeros rA, AB metietur. metiatur et sit z/. 

iam quoniam A numeroa FA, AB metitur, etiam 

reliquum Br metietar.') uerum etiam BA metitur. 

I 1 1 A ieitur AB, BF numeroa me- 

' s r . 

titur, qui mter se primi sunt; 

qaod fieri non potest. itaque 

imeros FA, AB nnllaB numerus metitur. ergo FA, 

IJ^S inter se primi Bunt. eadem de causa etiam AF, 

TB inter se primi sunt. FA igitur ad utrumuia AB 

\\Sr primus est. 

iam rursus FA, AB inter se primi siot; dico, etiam 
AB, Br inter se primoa esse. 

nam si AB, BF inter se jjrimi non snnt, numerua 
aliquis eos metietur. metiatur et sit jJ. et quoniam 
^ utrumque AB, BF metitur, etiam totum FA me- 
tietur.') uerum etiam AB metitur. ^ igitur FA, AB 
metitur, qui primi sunt inter se; quod tieri non potest. 
itaque numeroa AB, BF nultua numerus metietur, 
ergo AB, BP inter se primi sunt; quod erat demon- 
stvandum. 

XXIX. 
Quiuis numerua primus ad quemuis numerum, quem 
non metitur, primua eat. 

1) Neqne boc, ueque qno lin. 17 ntitur, nsquam apud 
Bucliaeiii demoitBtiatimi eet; pro axiomcbtiB ea habuit. cfr. 
CkTiuB 11 p. 12 nr. X et XII. 



28. la' BVp9, P m. rec. 24. utiavta^Kq. Ittoaiia 6 litt V. 
S6. 3» — iffri*} in rai. m. 1 p. fieTf^ ^< ^*"^- "^- ^- 



f 



248 rrorsEiflN £'. 

"fiffroj agmzos «ptftfiog 6 ^ xal toh S fi^ fii- 

TQSLXca' liya, Sn ot 5, A n^arot xqos aXX^^Xovg dGtv. 

Ei yoLQ fi^ iiai.1' oC B, A xpiaTOi. npdg ai.kiii.ovs, 

(t£TQ^06i, rig avTovg (tpt&ftoff. ytiTQtCtia 6 T. iJtEl h 

6 r Tov B {lETQst, 6 Sl A Toi" B ov [ittQit, S r aftt 

Tp A ovx i6ttv 6 avtog. xal iTttl 6 F Towg B, A 

(lEtQEt, xal xhv A UQa fieT(ifr itg^tov ourK p^ raw 

avra 6 avtog' ojuq ietlv aSvvatov. ovx aqa lovg 

B, A lietpijtfet tig aQi&nos. oC A, B aga Jtgiatoi Jttfog 

10 aXXilXovg tleCv oxeq iSei, Sst^ai.. 

A'. 

^Eav Svo dQi&fiol noXXanXaeittOavtB^ aX- \ 
AiJAows xoimel ttva, rov Si YevoftBvov i^ i 
twv (tEtQ^ Tig aQiatog aQi&fiiig, xal Sva tSm 
15 f'£ aQxVS tiitQ-^eei. 

^vo yuQ aQt&fiol oC A, B xoXXunXaeidaam 
aXX^^lovg tov P itoiEttaeav , tov Sl f iiBtQeCtm ■ 
irptatos aQt&iios o jd' Xsya, OTt 6 ^ sva tiov A^ 

flEtQEt. 

ao Tov yaQ A fiii fttTpftta)" xaC iatt itQmtog 6 ^' 

oC A, /i ttQa itQmzot xqos alXljXovg sleiv. xal offaxtg 
6 A Tov r fietQEt, ToffaiJTOi fiovdSts ioraeav iv Tp B. 
ijlEl ovv 6 ^ TOi' r ftstQst xaza tag iv t^ E p.ovat 
Sas, 6 z/ ttQa Tov E noXXanXaaideag tov P ji 

26 xev. aXla fiijv xal o A tov B itolXaTiXaatdaas i 
r itExoirjxEV Caos aQa iarlv 6 ix tav J, E tS i 



3. B, A] A, B p, 4. iieifrfids] -ob in ras. .,. ^.., 
TO) 93. ixfi] xal 6 r oiiii iiru ftOKne. iwfl ouv Vqs et c 
Mttf p. Ante iTiti ftdd. P m. rec. xaC. 6. -tm] e cort k 

£, jj] in TOB. <p; B aupra m. 1 Y. S. aviog ov6e fiovut Vggv 



k 



■ ELSMENTORUM LIBER VII. 249 

Sit numerua pnmus A et nnmerum B iie metia- 
tnr. dico, numeros B, A inter se primos esae, 

nam si numeri B, A inter ae pri- 

mi non aunt, numerus aliquis eo3 

metietur. metiatur numerua T. quo- 
niam T numerum B metitur, A 
autem h non metitur, r" et ^ iidem non sunt. et 
quoniam T numeros .8, A metitur, etiam numerum A, 
qui primus est, metitur, quamquam idem non est; 
quod fieri non potest, itaque nomeroa &, A nullua 
numenis metietur. ergo A, B inter se primi aant; 
quod erat demonstrandum. 

XXX. 

Si duo numeri inter se multiplicantea numerum 
aliquem effecerint, et numerum ex iia productum pri- 
mus aliquia numerus metitur, etiam alterutrum nume- 
rorum ab initio aumptorum metietur. 

A\ 1 Duo enim numeri A, S inter 

— — I se multiplicantes numerum T 

1 effieiant, et numerum T metiatur 

d primus aliquia numerua ^. di- 

H co, ^ alterutrum A^ B metiri. 

nam numerum A ne metiatur. et z/ primus est. 
itaque A, jJ ioter se primi sunt [prop, XXIX]- et 
quoties z/ numerum T metitur, tot unitates siut in E. 
iam quoniam A numerum T aecundum unitates nu- 
meri E metitur, erit ^ X E = T [def. 15], uerum 
etiam AxB •= T. itaque J x B = Ax B. quare 



250 rroiXEiiiN £'. 

tmv A, B. iaziv kqk wg o ^ irpos %ov ^, ovttos o 
B itQos tow E, 01 di /J, A XQatoi, oC S^ srQtBtoi 
xal ikaxtsxoi, ol di iXdxtOtoi (lEtQoiet rovg tov ttv- 
Tov loyov ^xovxas ieaxig o xe iisi^iov rgv (i.si^ove 
5 xal o ikaaeoiv xov ii.d<3Covtt, rovzicttv E re ijyov- 
liBvos xov tiYovftsvov xal 6 iTtoiitvog ror ixofttvoV 
6 ^ afftc xov B fisxQEi:, o[ioie>s rf^ <fei^Ofiev, ort xttl 
iav tbv B ft^ ftf^^^f ^^" -^ (iBtQ^^GEi, b /1 kqu tva 
xmv A, B (iEXQEf onsQ Idsi det^aL. 



10 Aa . 

"Aitag evv&ttog ttQi.9(i6s vxo «QOtov Ttvas 
ttQiQftov (iBXQEtxai. 

"EtSta evv&Bxog aQi&fios 6 A' liyei, Srt 6 j4 vnb 
XQmxov rtvog apt&(iov (tEXQsttat, 

15 'EiibI yaQ 0vv&bx6s iextv 6 A, f^stpijflat ng avrbv 
aQi&iiog. (tBXQBita, xttl iaxta o B, xal ei (liv «(xdtos 
iexiv b B, ysyovbs av eHri xb iaixax&^v. et dh evv- 
^■Exog, (lEXQ^^tfEt xig avxbv ttQi&(t6g. (iBtQfitm, xttl 
istta r. xttl intl 6 r tbv B (tEtQEtl, b Sl B xbv 

20 A (lEXQEi, xal b r aQa xbv A (kEZQEl. xal ei (iiv 
TCQiaxos iaxiv b F, yByavbg Sv eftj ro iattax&iv. el 
6i Svv&ETOs, (lETpjjffEt tig avtov ttQi&(i6s. roitcvxTis 
St} ytvo(i.ivr]s intOxi^eas lr[<p&ri0Etai rtg TtQatog «pi9- 
Itaq, og (iEtQ7]0Ei. Bi yUQ ov i.7j^&rjaEtai, fiETpijaovfft 



4 



3. Kol] om, p. of Si IXariCToi] supra P, in mg. m, 

1 V91. 4. fteijaiv idv] nig. ip (tvv in ms.). 6. rav] (prina) 
iu ras. tp. 8. B fifl in ras. p; (iij Bupra V m. 2. Post fittfi 
ras. 1 litt. p. 9. Seqni^r in BVpip alia demonatratio 

prop. XXX[ B. Theone addita; n. app. 10. ly' BVqi, P m. 
rec; 3.6' p, 14. fitiQficai P, oorr. m. 2. 17. Sijlov av 

fiiij to iiitoiiiEvov Theon (BVpq)). 20. fiFipti] (piius) 



■ ELEMENTOEUM LIBER VU. 251 

[prop. XIS] jd i A = B: E. uerum ^, A primi aunt, 
primi autem etiam miuimi sunt [prop. XSI], minimi 
antem eos, qui eandem rationem habent, aequaliter 
mefciuntur, maior maiorem et minor minorem [prop. 
XS], h, e. praecedens praecedentem et sequens se- 
quentem. itaque A numerum R metitur. similiter de- 
monatrabimus, A tiumerum, si B numerum non me- 
tiatur, numerum A metiri. ergo A alterutrum nu- 
f merorum A^ B metitur; quod erat demonstrandum. 

SXXI. 

Quemuis numerum compositum primus aliquis nu- 
Dierus metitur, 

Sit numerus compositus A. dico, primum aliquem 
llumerum numerum A metiri. 

nam quoniam A compositus est, uu- 
merus aliquis eum metietur. metiatur 
et sit B. et si B primus est, factum erit 
id, quod iuaai sumus.') sin composi- 
iliquis eum metietur. metiatur et ait T. 
et quoniam V numerum R metitur, et B numerum A 
metitur, etiam T numerum A metitur. et si T primus 
eat, factum erit, quod iussi aumus; sin compoaitus, 





1) Sc. primum nnmerum numerum A metientem inuenire. 
muamqn&m haec formula !□ problemata magiB oadit, id quod 
IL._ . .^-.. ,. 262, 12 nalet 



21. Siilav av ilii ro Jjiroufifvoc Theon (BVp^), 
' 22. PoBt nptfffios add, p: ftfipeiToi Kal ietia o T, ««1 kiiA 
pi T zov B fietptt a^ B xov A (letjsi, %al o V aea rov A 
HtTQft. 23. 3^] con: ei 8i T, Si pg;. 24. offl Bupra m, 
3 P. Poat (ttie^od add, Theon rov tiqo eatiioti, (huc uaque 
etiana P mg. ni. reo.) og xal lov A n,eti/-^att (BVpq;), et] 
oorr. ex ^ m. reo. P. ov] fi^ Angnst. 






252 ETOIXEIiiN £'. 

rov A apiQitbv axtigot api^noi, etv hsQog iripov 
ik«eetav l<izCv oaep letlv ddvvavov dv d^i&nots, 
ktj^i&^^attai xis UQa npmrog apLQfiog, oc^ it£T(^6ii 
rov wpo iavtov, os «al tov A ^ctpijffEC. 
& "Aitaq apa avv&sroq api&ftos uno «poitov Vivi^', 
aQi&iiov (LitQitrai,' OTteg iSfi Set^ai. 

//3'. 
"Anag dQt9[i6g ijtot JtpioTog iariv 
itpatov rtwog dpi&fiov [letQStrtti. 
"Eata d0i9fi6s 6 A' liyat, OTt 6 A r^rot iTfiroTog 
lativ 1] vito ngmtov tivos d(fi&fiov [iszpBft-at. 

Ei lisv ovv agmtog iativ 6 A, yiyovog av stiq x6 
imrax^iv. sl St avv&irog, (isrg^asi riff avrov jiQm- 
to? dptQfiog. 

B "Aitag aptt dgi&ftog ^rot itQiSrog iariv ^ vith itQta-^A 
Tov ttvisg ttpi9fiov fictpsitaf o;rfp Iffet Stt^tti. 

1-7 ■ 
'Agi&fiav So&ivtntv oaoaavovv svQstv to 
Haxiarovg rmv tov avrov koyov ixovvmv aS 
roZg. 

"Esraeav ot So&ivtsg onoaoiovv d^i&y^ol ot . 
B, F' Sai Sf) tvQstv rous iXaxCarovg rav tov < 
Aoyov ixovxatv xotg A, B, F. 

Ol A, B, r yaQ ^TOt irpraroi irpog dkXrjXovs sllA 



j IxsQOq V^ 



ifov BVpip. _ a. iexlv~\ ^iu) 



oiD. B, 3. npffl-roe apij^fios] Eupra m. 2 V, a^id^^oe K^atoi p, 

T. !«■ BV, P m. rec; Xi' p. 8. ^as P. 11. ^om V». 

12. ^'fyo^os] Pp, a^loii BVqs. 13. initai^iv} iji^avfietov 

Theon (BVp^). 17, Is' BV, P m. rec; iS' p. 19. to^s 



Theon (BVp^), 
aitoiis XQyovt Bp, 



Idyoue BVpq), 



> 



P ELEMENTORUM LIBER Vn. 253 

numems aliquis eum metietur. hac igitur ratiocinatione 
procedeute inuenietur primus aliquis numerus, qui 
metietur.^) nam si non inuenietur, infiniti numeri 
numerum A metientur, alter semper altero deinceps 
minores; quod in numeris fieri non potest itaque 
inuenietur primus aliquis numerus proxime anteceden- 
tem metiens, qui etiam numerum A metiatur. 

Ergo quemuis numerum compositum prtmus aliquis 
numerus metitur; quod erat demonstrandum. 

xxsn. 

Quiuis numerus aut primus est, aut primus uu- 
merus eum metitur. 

Sit numerus A. dico, uumerum A aut primum 

Iesse aut primum aliquem numenim eum metiri. 
^ iam ai primus est A, factum erit, quod iussi 
sumus; sin compositus, primus aliquis numerus 
eum metietur [prop. XXXI]. 

Ergo quiuis uumerus aut primus est, aut primus 
numerus eum metitur; quod erat demonstrandum. 

sxxm. 

Datis quotliliet numeris minimoa eorum, qui ean- 
dem ratiouem habent, inuenire, 

Dati fiint quotlibet uumeri A, B, T. oportet igi- 
tur minimoa eorum iuucnire, qui eandem rationem 
kbeant ac A, B, F. 

B, r enim aut iuter se primi sunt aut non 

1) Sc, numeium pt&ecedeutem et ea de canBa DUineruiu 
t (cfr. lin. 4). et puto, haec audivi poaae. etai fieri poleet, 
it haec uerba in P mero errore ob ojioiozfltviov esciderint. 




264 ETOiSEiaN t'- 

^ ov. ei (liv ovv oC A. B, F Ttffmtoi «qos uHJIt^ovs 
Bioiv, ika%i9%oC tiei xmv rov avrov i,6yov i%6vz^v 
avToti. 

ti &\ oij, tikri<p&ai Tfov A, B, F to ftiyiijTov xoi- 
5 vov fiSTQOv 6 ^, xal oaaxig 6 .^ txa6Tov zmv ji, B, F 
(terQEt, ToSavzai (lOvaSdg iezaOav iv ixaSTa zmv E, 
Z, H. xttl fXKffros apc rtov E, Z, H exaifTov tmv 
A, B, r (lETpet xaxci t«s iv za // ftowadaff. ot E, 
Z, H apu tovg A, B, F i««xis fttrpoijfftv o^ £!, Z, H 

10 Spa rots A, B, V iv roi aijrp A.6ya eiffiv. Kiya tfij, 
ort xal ilaxtiSToi. ii yaQ jtJj eloiv ot E, Z, H iXa- 
%ietot. riav rov avTov XCyov ijovrtav zotg A, B, F. 
BOovTat [zivsg} %mv E, Z, H iXaeooves «pt^jtol iv 
rra aurp loyui SvTeg toEs A, S, F. Setioeav ol S, 

16 K, A' iedxtg aga 6 & rou A fiErgat xal ixaTe^oi 
T«tv K, A BxdteQQv Twv B, F. oedxig 8% b & riiv A 
(lETQti:, tueavtai itovddss letoGav iv rp M"' xal ixd- 
TSQog aQa tmv K, A sxuteqov rrow B, F fteT^et xata 
r«s iv rto M fiovddag, xal i%el o & titv A (lecQel 

20 xata ras iv rp M [lOvdSag, xul 6 M aQa row A (w- 
rpEl xat« rag iv rio & novdSttg. dia ta avta S^ o 
M xal ixdTiQov rwv B, F ^Er^ft xara rag iv ixa- 
rEpoj rrow K, A jiovddag' 6 M apa loug A, B, F 
(istQst. xal ijtel 6 & xov A (teTQst xara tag iv 

26 M fiovdSttg, 6 & UQa tov M aoXlttaXaaidaag t6v 



, r 



6. Iv] om. P. 7. Haatos] cnaarov p. 10. «rs] corr. 
zai ra. rec. P. Bfo^ V?!. 11. ««t'] kcI oC p. 12. Tofs] 
T. es lot m. 1 P. 13. rivtt] om. P. 16. B, P] F, B 

T. es A, B ia. i p. ei] *n? 18. rav B, T] t=- "— 

p. 30. ^] e p. 21. «al M Vqi. 



B, rj TC^ ^ 



ELEMENTORUM LIBER Vn. 265 

primi. iam si Ay B, F inter se primi sunt^ minimi 
sunt eorum, qui eandem rationem habent [prop. XXI]. 

sin minus, sumatur numerorum 
A, By r maxima mensura communis 

r ^ ^ [prop* 111] 0> ®* quoties z/ singu- 
n los numeros A^ By T metitur, tot 



JB 



IT 



lii 



M E- 



Kh Z» 



unitates sint in singulis E, Z, H. 
quare etiam singuli E, Z, H singu- 
los Aj B, r secundum unitates numeri 
^ metiuntur [prop. XV]. itaque E, Z, H numeros Ay 5, 
r aequaliter metiuntur. itaque Ey Z, H et Ay B, F in 
eadem ratione sunt [def. 20]. iam dico, E, Z, H 
etiam minimos esse. nam si E, Z, H minimi non 
sunt eorum, qui eandem rationem habent ac A, B, F, 
erunt numeri numeris E, Z, H minores, qui in eadem 
ratione sint ac A, B, jT, sint &, K, A, itaque & 
numerum A et uterque K, A utrumque B, F aequa- 
liter metitur. quoties autem & numerum A metitur, 
tot unitates sint in M. quare etiam uterque K, A 
utrumque B, V secundum unitates numeri M metitur. 
et quoniam ® numerum A secundum unitates numeri 
M metitur, etiam M numerum A secundum unitates 
numeri ® metitur [prop. XV]. eadem de causa M 
etiam utrumque B, T secundum unitates utriusque 
K, A metitur. M igitur numeros A, B, T metitur. 
et quoniam ® numerum A secundum unitates numeri 
M metitur, erit ® X M = A [def. 15]. eadem de 



1) Cum sroQKTfta prop. S spurium sit, Euclides tacite 
eam ad quotlibet numeros transtulit; cfr. p. 269 not. 



256 ETOIXEIiiN £'. 

jitJtoCrixtv. Sia ra avxa dij xal 6 E Tov ^ TcoXXa- 
nXaeLteaas zov A XBTCoirfxtv. i^aos uqa ietlv 6 ix tmv 
E, ^ tp ix ratv &, M. iauv apa as 6 E x^og toc 
&, ourojs o M srpog rov A. fiti^atv dh 6 E tov ©' 
6 fifilmv «pa xal 6 M tot; z/. xal iierQil rovg A, B, P 
ow£p iiJtlv aSvvaToV vnoxeiTai yaQ o d rt5v A, B, T 
to (liyiatov xotvov fiit^ov. ovx «pa leovtai ttvts 
rmv E, Z, If ikauaovss aQi&fiol iv ta «vvp loya 
ovTSg tots j4, B, r. 61 E, Z, H apa ika-g^L0toC tiot 

10 Ttni' tov owrov koyov ixovtaiv toig -^, B, /'■ omfp 
idst Stt^ai. 

f.d'. 
^vo dpi&iimv So&dvtmv svQstv, Sv iXd- 
XiOrov fi,£Tgovaiv aQi&fiov. 

16 "EermOav ot SoQivtts Svo apt^fiol of A, B" 8il 
Sij EVQetv, ov ixdxiotov fiBtQovaiv apiftfioV. 

OC A, B yag iJTOi jrpcirot srpog (iAAjjious tlelv §} 
ov. ietmeav JtgottQov ol A, B nq^toi Jtpoj aAlij' 
lovq, xal 6 A Toi' & itohl.u.Ttl.a.eideas rov T Ttoitito' 

20 Ka\ 6 B uQtt rov A nollaTtkaeideas tov P ntjtoiijxtv. 
of A, B aga tow P fittQoveiv. Xiyat JiJ, 
iXdxietov. ti yap [tri, ftttQ-qeovsC riva aQiS^fiov 
A, B iXdeeova ovta tov P. ftttQeitoeav toi' ^. xti 
oedxis 6 A toi' .d (ittQtt, roeavtai (lovdSeg iatfoeav 

25 iv TM E, oedxiq Se 6 B rov .d (lerQtt, toeavrai fto- 
vdStg ieraeav iv ta Z" o (itv A apa tov E aoXXa- 

1. wtuaiijxe Yip. iiaia — 2: mxoirj-xev] om. |j, 8. o»ks 
iv tffl auim loym p. 9. eletv P. 12. IS^ BV,_ P m. rec; 
3.£' p. 15. diio (i^iO^iol of do&ivTtq p. 16. aptdfiof] oni. \<p. 
19. tow r — 20: TtoUanXaeidaas] rag. m. 2 B. 20. aja] 

comp, Bupra V, iu tf, 21. KOil oi P. ftfieovfli V9. 32. 



KtV.. 



ELEMENTOKUM LIBER \^I. 257 

cauaa erit etiam Ex./1 ~ A. itaque Ex/l = &xM. 
quare erit [prop. XIS] E:® = M'.^. uerum B > @, 
qiiare etiaiii M> z/ [prop. XIII. V, 14]. et M nu- 
meroB A, B, F metitur; quod fieri non potest. nam 
auppositum est, ^ maximam mensuram communem 
esse nmnerorum A, B, r. itaque non erunt nuraeri 
Eumeris E, Z, H minoreSj qui in eadem ratione sint 
ac j4, B, r. ergo E, Z, H minimi simt eorum, qui 
eandem rationem habent ac A, B, F; quod erat de- 
monstrandum. 

XXSIV. 

Datis duobus numeria, quem miiiimum raetiuntur 
numerum, inuenire. 

Sint duo uumeri dati A, B. oportet igitur, quem 
minimum metiuntur numerum, inuenire. 

A, B enim aut inter ee primi suut aut non primi. 
priuB A, B inter se primi sint, et ait ^4 X B = F. 
quare etiam B X.A = F [prop. XVI]. itaque A, B 
namerum V metiuntur. iam. dico, eos eum etiam 

minimnm metiri. nam si miuus, A, B numerum ali- H 

q^oem numero V minorem metientur. metiantur nu- 1 

memm zJ. et quoties A numerum ^ metitur, tot 
nnitates sint in E, quoties autem B numerum A 
metitur, tot unitates sint in Z. itaque erit AxE = tS, 





258 ETOIXEIHN £'. 

xXaataeag zov ^ jtfjto^jjxEv, 6 d£ 5 rhv 
3ti.aaitteag xbv ^ nBnofnjxev tSog aga iOTlv 6 ix tav 

A, E rm bx zav B, Z. Sotiv apa lag o A itpo; ibv 

B, oiJitos 6 Z jrpos Toi' E. of tf^ A, B nptoroi, oC 
& Si jrptOTOf xal ^Aaj^tffroi, oi Se ikayioxot fistpovai rovs 

Tov avrov X6yov £y_ovxag isaxig o xb ftECt^oiv tov (isi- 
%ova xal ikdeetov zov dXaaaova' 6 B «pa rbv E 
(lEXQtt, ag ijtofiivos ETiofitvov. xal ejceI 6 ji rovg B, E 
itokXaaXaOtaOag xovg F, z/ nEXoii^xEv, iartv apa ag 

10 6 -8 jrpds Tov E, ovrmg o J' «pos rov /J, (ter^Bl Si 
b B tov E- [lETpEt a^a xal 6 F xbv ^ o ^EC^av 
thv iXaaaova' ojisp iozlv advwaToi'. ovx UQa oC A, 
B (iet^ovOt ttva dQi&(iov iXdeGova ovta tou r, 6 P 
UQa iXdxtexog mv vno rav A, B fiEtQEttai. 

15 Mjj iaraeav Si} oC A, B •xqatot irpog dXX^lovq, 
xal BlXrifpQ-coeav iXu^tatoi apiftftoi rmv rov aurdw X6- 
yov ixovrav xotg A, B ot Z, E' teog aga ierlv o 
ix xav A, E rra Ix rav B, Z. xal h A rov E jroAict- 
TiXaaidoag xov F TCoiEtxa' xal b B aga rbv Z xoXXa- 

20 aXaeiaeag lov Z* Tisnoiijxav of A, B «pa Toi/ T (ts- 
TpoiJffti'. Xiya S^, on xal iXdxtSxov. sl yap (i^, 
(isrp^^eovei tiva difi&(ibv oC A, B iXdeeova ovra rov r. 
(istQECzmeav tbv /i. xaX hadxtg (ilv b A xbv ^ (ie- 
tffEl, roaaiJrai. (lovddEg setcoaav iv rm H, 6a«xis Si 

25 6 B rdv A jiitQEi, xoettVTat (lovdSag iaxaaav iv Tt3 8. 
6 ^^f A aga xbv H aollanXaeidaag tbv A xtitoi^XEVf 
6 Si B tbv & xoXlanXaatdeag tbv /1 itETCoCrpisv. 



3. A'^ (prinB) corr. ex d V. 6. fKrpoiJinf B. 9. T, 
T poBtea inaeTt. m. 1 p, poet ti 1 Htt. cras. 11. &^ti\ 
Kpa p. TOj' ^] T^* J P. 13. ftMpTJoodoi* P. Post to6 r 
add. Theoo: oin;» of A, B »e"^0' ^Q^S oil^loi'? moi» (BVp% 






H ELEMEKTORUU LIBER VU. 259 

B X Z «= z/ [def. 15]. itaque ^ X £ = B X Z. 
quare erit A : B <!= Z : E [prop. XIX]. uerum A, B 
primi suut, primi autein etiam minimi sunt [prop. XXI], 
minimi autem eos, qui eandem rationem habent, aequa- 
liter metiuntur, maior raaiorem et minor minorem 
[prop. XX]. itaque S numerum E metitur, ut sequena 
sequentem. et quoniam A numeros B, E multiplicans 
numeros P, ^ effecit, erit B : E ■= F: J [prop. XVII]. 
uerum B numerum E metitur. quare etiam r nu- 
merum ^ metitur [def, 20], maior minorem; quod 
fieri non poteat. itaque A, B nullura numerum nu- 
mero F minorem metiuntur. ergo F numerum mini- 
mum metiuntur A^ B. 

Ne sint igitur A, B inter ae primi, et aumantur 

I jI B Z, E minimi eorum, qui eandem 
ratiouem .habent ac A, B [prop. 
j ,2 I lExXXin]. itaquev^xB^BxZ 
•i r ' [prop. XIS]. et sit ^ X E = T. 
' 1** itaque etiam B X Z = F, quare 
'■'" '^ I— iS j4^ s numerum F metiuntur. iam 

dico, eos eum etiara miuiraura metiri. nam si 
miuus, A, B numerum aliquem numero P minorem 
metientur, metiantur numerura ^. et quoties A 
numerura z/ metitur, tot imitates aint in H, quoties 
autem B numerum jJ metitur, tot unitatea aint in ©. 
itaque A X H ^ J, B X & = .J [def. 15]. quare 

P B. rec.) 15. 8ij~\ /l{ p. 17. Z, E] corr. m £, Z V. 

19. Tov r — mAlaitJicieiaaas] mg. m. 1 p, noitJroj — 20; 

lov r] nig. tp. 30. «molijHB p. jiixtfovai Vw. 22. fiE- 

T9^«aii«>i' FB, (lexgijaovai 3^ p, 24. H, oauHit — 3S: iv 

rio] om, p. 26. J] cort. ex A p. 27. o 9i B — wsnoiij- 
niv] om. p, 

_ n* 



258 STOIXEKN f. 

7cXa6id6ag tov /1 nsnoirjxev, 6 dl B thv Z 
ytXa6id6a$ rov z/ TtcxoiTjxsv' t6og aga i&tiv 6 
Aj E TcS ix, tmv jB, Z. £6tiv aqa mg 6 A at^ 
Bj ovtfog 6 Z TtQog tbv E. ot S^k A^ B ^w 
6 dh iCQmtOL xal ika%iAStoL^ ot 8% ika%L6toL iietfov 
TOi/ avtbv X6yov i%ovtag l6axig o ts futf/n» r 
iova xal 6 ika66ayv tbv ikd66ova* 6 B ofa 
[istQStj mg s^oiisvog ixofisvov. xal insl 6 A ro» 
nokla7tXa6Ld6ag tovg F, A TtSTtoirixsVj l6tiP f 

10 6 B XQog tbv E, ovtfog 6 F Jtgbg tbv A, fLf 
6 B tbv E' (istQst uQa xal 6 F tbv A h 
tbv iXd66ova' otcsq i6tlv ddvvatov. ovx &Qt 
B [istQ0v6i tiva aQLd^^ibv iXd66ova ovta tov j 
aga iXd%L6tog mv vjtb tSv A^ B (istQsttaL, 

15 Mfi i6t(o6av Sri ot Ay B TtQmtov Ttgbg dX/ 
xal slkrjtpd^m^av ikd%L6xoL aQLd^iiol tmv tbv avy 
yov i%6vt(ov totg A, B ot Z, E' t6og aqa t 
ix tmv A^ E tm ix rmv B^ Z. xal 6 A tbv E 
nXa6Ld6ag tbv F TtoLsCtm ' xal 6 B aQa tbv Z 

20 nXa6Ld6ag tbv F jtsTCOLTixsv' ot A, B aQa t(n 
tQ0v6LV. ksyco drjj oxl xal iKd%L6xov, sl yi 
(istQ^6ov6i tLva aQLd^^ibv ot Aj B iKd66ova ovta 
(istQsix(o6av tbv A. xal b^dxLg (ilv 6 A tbv 
tQstj to6avtaL (lovddsg s6t(o0av iv tm H, 6tfi 

25 6 jB tbv A (istQst^ to6avtaL (lovddsg i6tm6av if 
6 (ilv A aQa tbv H 7Co^.Xa7cka6Ld6ag tbv A srcst 
6 dh B tbv 7CokXa7cXa6Ld6ag tbv A TCSTCoiijxst 



S. A] (prius) corr. ex ^ V. 5. (isTQOvaiv B. 9 
r poBtea insert. m. 1 p, post z/ 1 litt. eras. 11. 
uQa p. tov Jl TT^v d F. 13. fiexQTjaovaiv P. Pof 
add. Theon: otav ot A, B jcqoozol nQog dXXi^Xovg (Dtfty 



' B, St^^w 



260 STOIXEiaN £ . 

apa ^OtIv o ix Ttov A, H toj ix Ttoi' . 
«pa wq b A 3ipog row B, ourwj o © ^Qog; TOf H. 
i»s di 6 .^ ■KQoq xov B, ovzag 6 Z wpog toj/ £■ xal 
OJS apa o Z npos tov E, ovtcos o @ XQog zi.v H. ol 
f> dh Z, E ikdxiaToi, ot 6% iXapgzoi (lerpovat rovg xov 
avTOV loyov Ixovtas iaaxig o te [lii^av thv (tei^ova 
xaX 6 ildattav tov iXdaaova' o E «pa rov H (ttrQti. 
xal iml o A Towi; E, H noXlazlatftaeag zoifg F, A 
Tttxoi^^xev, letiv aga tog 6 E wpo? tbv H, ovTfag 6 
10 r itQog tov ^. 6 Si E zbv H fiitQet- xal 6 r apa 
rov ^ fiBTQEi 6 fisi^av rov ildaeava' Sjieq ietlv 
aSvvatov. ovx aqa ot j4, B (letff^eovei tiva d(fi&iu 
ildeeova ovta tov P. l F at/a ii.d%tetoe mv i 
Tiav A, B ftirgaiTai' Smp l^et Sti%ai. 

16 M'- 

'Ettv Svo dpi&ftol dgi&fiov Tiva [lezgweipt 
xal 6 ii.iixteTos vTi (tvrcDV (letQovfievos to 

aVTOV ^EtpjJffEt. m 

^vo yiiQ aQi&^ol ot A, B aQt^fiov tiva rbv PM 
20 (tszQeitaeav, ikd%iGtov &e tov E' i,iya, Sti tcal ' 
tbv r.A (letQst. 

El ydp ov ftitQet 6 E tbv T^d, b E tbv jdZ, j ._ 

Tpfov XemiTm iamov ildeeova tbv FZ. xal intel ot 

j4, B tbv E fisTQOveiv, 6 61 E tov /JZ (ietQsE, xal 

25 ot A, B aQtt tbv AZ (letQ^^eoveiv. ^GtpouiTi Si xal 

2. tos] inaert. m. 1 p. Hl in rag, ip, _ 3, oiltojs 6 Z 

n^og tov £] tDg. 9. Post E add. P: all' mg A Tcgot t6p 
S, ovztoe o 8 agbs tov H; deL m. rec. xrI tBg apa] ^ottv 
Si/a tos p. 4. 2] P, cotT. rn. reo. E] H P, corr. m. 
reo. 0] Z P, corr. m. reo. H] E P, corr. m. rec. S. 
Tovs] lov p. E, H] H, E B. 18. fi£t0^«ov«v B. tS. 



■ ELEMENTORUM LIBER Vn. 261 

A>CH=BX&. itaque A:B = @:H [prop. XIX]. 
uerum A : B = Z : E. itaque etiam Z : E = @ : H. 
uerum Z, E minimi sunt, miuimi autem eos, qui 
eaudem rationem habent, aequaliter metiuntur, maior 
maiorem et minor minorem [prop. XX]. itaque E 
numerum H metitur. et quoniam A numeros E, H 
multiplicans nnmeros F, A effecit, 'erit E: H = T: A 
[prop. XVII]. uerum E numerum H metitur. quare 
etiam F numerum A metitur [def. 20] maior mino- 
rem; quod fieri non potest. itaque A, B nuUum nu- 
merum nnmero f minorem metiuntur. ergo F nu- 
merum minimum metiuntur A, B; quod erat demon- 
atrandum. 

XX5V. 

Si duo numeri numerum aliquem metiuntur, etiam 
[jliiem minimum metiuntur numerum, eundem metietur. 
Duo enim numeri A, B na- 

__, - merum aliquem FA metiantur, 

Z .minimum autem E numerum. 

dico, etiam E numerum nume- 

i r^d metiri. 

Nam si E numerum F/i non metitur, E nume- 
rum ^Z metiena relinquat ae minorem FZ. et quo- 
niam A, B numerum E metiuntur, E autem numerum 
^Z metitur, etiam A, B numerum ^tfZ metieutur. 



HFvtb] om. Vip. 16. i£' BV, P m. rec, Iri' p. 16. Poat 

^plav roB. 3 litt. BV, iinf^aaiai p, fictpmoi PYtp. 20, itai] 

"topra m. 1 P. 32. «;] (jj) Aiigiiat FJ] T B. JZ] 

ZJ p, r^ V in TBS,, <f. 26, iurf^aavtiv. nEZQovai] -at 

fUT^ov- add. m. 3 B; (iBtg^aovai fiEtQoiai Vpv; iitTgovatv. 

fMT^vai P. 



l 



iXdeaovcc ovza 
afia ov fMTpEt 



STOlXEIflN t'. 

'.al Aomoi' UQa zov FZ fiev^aov^ 
Tov E' OZEQ iarlv ttSvvaro-v. ovn 
t E risv V^' (lEtQei: aga' oxbq £i et 



TQiav dQi.&i.iiov do9svtav ei^QEZv, ov iJ!tP 
jjKjjroi/ (teiQOvaiv aptS-ftdv. 

"EaTaeav oC So&ivtes tsfEtg apidftoi of j4, B, P 
SeI Stj evpElv, ov ikaxiOtov jiBtQOvaiv api&ft.6v. 
Ell^(p&ia yag vxo Svo tav A, B iXKXiatos fu- 

t^OVllBVOS ^. O dij r tOV A ^TOt flftpft ■q ov (M- 

xqBl. (itzQ6(ta Kpotspoi'. fiEtQovai Sl xal oC A, B 
zov A' ot A, B, r apa tov A y.ttQOvGi.v. Xiyca 4^, 
oxi KoX iXa%iaxov. eI y«p fiij, fterpjjffovefi' [tiva] 
15 «piftftov ot A, B, r ikaaaovu ovta zov ^. liEtQB^- 
tioaav tbv B. imtl ot A^ B, F tbv E [iBtQOveiv, xal 
ot A, B «p« zbv E ^etpovaiv. xal o iKdxtatos 
vxb tmv A, B (iBz^ovfiEvog \tbv E] fi^rpijaet. H 
^tfftos Sl vnb Ttav A, B (istQOVfiEvog isttv o z/* 
«pa tov E fterp^dft o [iBi^civ tov ii.daaova' 
iezlv dSvvatov. ovx UQa oC A, B, F (iBtQ^aovaC riva 
ctQi&fibv Hdoaova ovra zov /i' of A, B, F Sga iXd 
Xtatov TOT' A iisTQovaiv. 

Mi^ jiEZQEiza 3}] xdkiv h F zbv A, xal fMiji 



'V4« 



5, Iq' BV, 19' p. 9, ii.STQ^BOvaiv P. 10. iiSji] 

ras. ip. 11. d^] 8i P. 13. «sa A, B, F Vip. /UEipoSffi 
Vpqi, fi.STe-^eoveiv P. tfij] om. Yip. 14. ficiQ:^aovai V 

et corr. ex lifzgfiaovoi <p. ziva} ont. Fp. 15. di/t^iiov] 

ora. p. iXdoBova] xiva dei^nov iXdxiovet p. 16. inel ovv 
Vtp. fterpoiai PVpip. 17. /iczQiiaovaiv P et comp. p; pt- 
movai Yip. 18. rov E] om. P. 20, fitr(nj(i£i] comp. p, in 
tiM. qj. 21. r] inaert. postea rp. fifr^ifoavat* B, fiet^ovoi V^ 



ELESIENTOEUM LIBER VU. 263 

uerum etiam totum T^ metiuutur, quare etiam reli- 
quum rZ metientur numero E minorem; quod fieri 
nou potest. itaque fieri non potest, ut E numerum 
r^ uon metiatur. ergo metitur; quod erat demon- 
strandum. 

XXKVI. 
Datis tribus numeris, quem minimum metiuntur 
numerum, inuenire. 

Sint tres numeri dati A, B, 
r. oportet igitur, quem mini- 
mum metiuntur numerum, iu- 
— — ir 

uemre. 

sumatur enim, quem duo 
numeri ^, B minimnm meti- 
untur, J [prop, XXXIV]. F igitur numerum ^ 
aut metitur aut non nietitur. metiatur prius. uerum 
etiam ji, B numerum ^ metiuutur, itaque ^, B, F 
numerum ^ metiuntur, iam dico, eos eum etiam 
minimum metiri. nam si minus, A, B, F numerum 
numero ^ minorem metientur. metiantur numerum 
E. quoniam A, B, F numerum E metiuntur, etiam 
j1, B numerum E metientur. quare etiam, quem 
minimum metiuntur ^4, B, numerum E metietur 
[prop. XXXV]. quem autem A, B minimum meti- 
untnr, est ^. A igitur numerum E metitur, maior 
miuorem; quod fieri non potest. itaque A, B, P 
nullum numerum numero ^ minorem metientur. ergo 
A, B, r numerum .d miuimum metiuutur. 

rursus ne metiatur F numernm ^, et aumatur, 

sa. r] om. F. 83. iitxe^ooviiiv F, comp. p; fi(Tpavn V^. 
S4. t^] di p. 



l 



264 



rroisEiiiN f. 




vitb riSv r, jJ iXdziOTos liizQOvtitvos api&itos t 
ixel ol. A, B Tov J iiEZQOvetv, A de jJ xov E (is- 
tgst, xal oi A, B aqa. Toi/ E (isTQovOiv. fiezQsi Si 
xal b r [xov E- xal] oH A, B, F uqu zov E fier^ovsiv. 
6 Xfya S^, OT-t xal sX6%iezov. ei y«Q ^tJ, ^erpTJffowffi' 
rtva o( A, B, V ikaeaova ovra rov E, (istQsitcoOav 
lof Z. inal ol A, B, F xov Z (ierQovttiv, xal o[ A, fl 
«pa roc Z fiBT^avatv xal 6 iXaxiaro^ Kp« vn^ rav 
A, B (lerQovp-svog tor Z (ietq^^Ssi. iXdparos 8% wii- 

10 tav A, B (i£tQov(isv6s iattv b ^d' h A aptx tov Z 
litTQst. (leTQst Si Xttl 6 r Toi' Z" of A, r apa lov 
Z (istQovOiv Sare xal 6 iXaxtatog vxb rmv ^, T 
^tQov^Bvos thv Z (leZQ^^aet. o Sh iXa^iatos vnb tav 
r, A (tetQov(ifv6s iariv o E- b E «pa z6v Z (lEtQel 

16 6 ftEi^cjv rov iXdsaova' Srccp ^ffilv «tfwvarov. 
«po; of ^, B, r fiErp^ffouflir' riva KptS-fiov iXdaaoi 
ovra rotj E. b E ap« iXdxtarog av vnb tmv j4, BJ^ 
(letQeitttf ojtep 6'dft ^Er^ort. 

h: 

20 '£av api^fio? virii tivos dQt&(t.ov iievQ^tm 
o ftfrpovfifvog 6fit6vv(iov (liQOg ^|ei Tp 
rpowvrt. 

'AQt&ltbs yiiQ b A ino tivos uQt&fiov rov S j 



1. apt^iudf] om. p. 2. ittigoviii Yip. ^] corr. 
m. 2. 3. Post B io p in. 3 inflert. r. pfip^ooinri* P, , 
Tpoiiiri Vy, comp. p. fterpff — 4; fierpoiioii'] ora. p. 1. 
lo»' E. nai] om. P. T] supra m. 2 V. fifipjjoouoi P, fie- 
ipoiJot Vqs. 5. 3jj] om. Vip. fisrp^oDTJon' B, comp, p; 

liexQovai Vip. 6. iiva] om. p. ii*c iXatTova dgLS^iidv ov- 
icE p. T. fiFipotJoii' , *ttl 01 ^, £ upa lov Z] mg. cp (^e- 

ttovot). attsavai Vp. imi oC /1, fi aQa xov Z f(Eipocoi.v] 
mg. m. 2 V. 8. (ifipoiioi*] iitTs^aovnt V, comp. p, in riLB. ^p. 



■ ELEMENTORUM LIBER vn. 265 

quem r, ^ minimum metiuntur numerum, B [prop. 

XJLX.IV]. quouiam ji, B numerum ^ metiuntur, et 

jj numerum E metitur, etiam ^, 3 

numerum E metiuntur. uerum etiam 

r numerum E metitur. itaque ji, B, F 

numerum E metiuntur. iam dico, 

E eos eum etiam minimum metiri. nam 

ai minua, ^, B, F numerum aliquem 

minorem numero E mefcientur, me- 

tiantur numerum Z. quoniam J, B, F numerum Z 

metiuntur, etiam ^, B numerum Z metiuntur. quare 

etiam, quem minimum metiuntur .4, B, numerum 

Z metietur [prop. XXXV]. uerum quem minimum 

metiuntur J, B, est z/. J igitur numerum Z meti- 

tur. uerum etiam T numerum Z metitur. itaque 

jd, r numerum Z metiuntur. quare etiam quem mi- 

nimum metiuntur ^, r, numerum Z metietur [id.]. 

uerum quem minimum metiuntur T, ^, est E. itaque 

E numerum Z metitur, maior minorem; quod fieri 

non potest. itaque nuraeri J, B, F nullum numerum 

numero E minorem metientur. ergo E minimus est, 

qnem ^, B, r metiuiitur; quod erat demonatrandum. 



^n, TOv Z — 10: tiftl/ovfxevos] om. p. 12. fidpijaouoi p. inote] 

"Dm. p. Sga V7i6 p. r, j p. 14. r, j] Ppi d, r 

BYip. 16. B) om. p. fiEipTJoovDi] PB. oomp. p; iu- 

Tpotioi V qs. flafcova tou E o»io p. 19. 19'' B (post add. 
m. 1, nt posthao siiepiuB), V, P m. reo., fi' p. 20. fieTpiiTci ip. 



xxxvn. 

Si numerum numerus aliquis metitur, ia, quem 
Bietitur, partem habebit a metiente deuominatam. 
Numerum enim A numerus aliquta B metiatur. 



266 2T0IXEI2N E'. 

TQcia^a ■ Kiya , oti b A oiitivvftov fi^^s Sj^ 
ra B. 

'Oaaxis yuQ 6 B roi' j4 (lETpst, roffai5T«£ (lovditi 
iOvsaeav iv tm J'. ijCBl 6 B tov j4 [itTffit xara rag 
b iv ra r fiovttdas, (urptt di xul ^ ^ fiovccg zbv T 
KQiQ^yiOV xazu tccs iv avrw (lovaSag, !aaxig a.Qtt ■>} J 
jiofas Tov r uQi&nov [UTQet xal o B tov A. ivaX- 
%.k% Bpa iattxig r] A fiovag t6i' B cJpi^fiow fi^TQsZ xal 
6 r Tov A' o «Qtt [lipos iarlv ij ^ ftoraj roi) B 

10 ttQi&[iov, t6 awTO (liQog ioTl xal o F toij A, ii Se /S 
(lovicg Tov B Kgt&fiov (liQog iorlv 6iitavv(i.ov avra' 
xttl 6 r ttQu Tou A liiffog iarlv hfiiawftov xa B. 
marE 6 A (liQos ix^i rov F bndvvjiov ovxa x^ B' 
oTtcQ iSei Stt^tti. 

16 ^»?'' 

'Eccv ttQt&fios fiipog £xV ortoiJj', vTtb ofta- 
vviiov aQi.&fiov fifrpij&iffferat rp. fiEQEt. 

'AQi&fibg yttQ A (liQos ixi^ca* ottovv xbv B, xat 
rra B (liQEi hy^mvvfios iara [lipiftfios] 6 F' kiya, St» 

20 6 r" ror A fifTpEf. 

'ExeI yttQ b B rou A (liQos ieriv 6(iiavv(tov t6 
r, iari dl xal ij A ftovAg rou T (ligog ofu-oii/vftiw 
aur^, o «pa (liQog iUTlv ij A fiovceg rot' f KQt&ftov, te 
avr6 (liQog iezl xal 6 B zov A' ieaxis «pa "^ A fio- 

S5 "as Tov r aQi&(i6v ftETQEt xal 6 B zhv A. ivaXlk\ 



2. tb] corr. ei to m. 2 T. 4. ira] om. if. rj eras, V- 
10. fiEPp^] nig, tf. 13. F] in ra:S, tp. Ofimvvfioi' cov T p. 

ontt] of- BQpra m, 1 P; om. p. 16, fi' BV, P m, rec,: lu»' p. 
16- ijTo] m. a B. 18. td*] rd P?i, et e corr, V. 19, 

Ofiojnjfio* p. ffipi*pde] om. Pp. 20. A~\ corr. ei B p ro. 1 1 
31, la\iv\iex\ %a£Y<p. 22. iariv PB, comp. p, as. /ti- 



H ELEMENTORUM LIBER VU. 267 

dico, numeram A partem habiturum esae a numero 
B denominatam. 

I 1 Nam qaotieB B aumermn A me- 

1 1 B titur, tot sint uuitatea in T. quoniam 

I 1 r -B numerum A secundum uuitates 

\—\A uumeri F metitur, et etiam unitas /J 

numerum T sectmdum miitates eius metitur, A unitas 
numerum F et B numerum A aequaliter metitur, itaque 
permutatim A unitas uumerum S et f numerum A 
aequaliter metitur [prop, XV]. itaque quae pars est 
/t unitas numeri B, eadem pars est etiam T uu- 
meri A. uerum A unitaa numeri B pars est ab 
ipso denominata. ergo etiam T numeri A pars est 
a B denominata. quare A partem habet F a B 
denominatam; quod erat demonstrandum, 

xxrvni. 

Si numerus partem quamlibet babet, numerus a 
parte denominatus enm metietur. 

_ jn- — — 1 Numerus enim A partem 

^k I 1£ quamlibet habeat B, et a parte 

^M ) 1 r B denominatus ait T. dico, nu- 

^K- I — -i^ merum T numerum A metiri. 

^K Nam quoniam B numeri A pars cst a T denomi- 

^raata, et etiam /i uuitas pars est numeri T ab ipso 

denominata, quae pars est A unitas uumeri T, eadem 

pars est etiam B numeri A. itaque A unitaa nume- 

rum r et B numerum A aequaliter metitur. itaque 



268 ETOiSEraN :'. 

'AQi&iibv eiipitv, og iAaxtOtog (ov i^ti xe 

5 Jodt'vr« flBQTl. 

"Eera za So&dvza fiiptj ra A, B, F' Set d-i] opi&- 

(ibv tiigsCv, og iXd%i.azos av f|f( t« A, B, F [U^. 

"Eaiaecsv yap rolg A, B, F [iBp£ai.v oinowitoi dgi^- 

/lol 01 z/, E, Z, xal iU^qi&fo vjtb tav ^J, E, Z it.dfy 

10 OTOg fMipowfWvos ugi&(i6g o H. 

'O H aga bfuovviia (teg^ cxn rolg /d, E, Z. toIs 
S\ A, E, Z b^mvv(ta ftBQri i<fr:l tK A, B, V- b HaQa 
iy^ii ra A, B, F jii^ri. liya dij, ort xal ild^ta-cos ^v. 
ti yaQ lii), iOzat ris rov H ildcacov dgi^fiog, og I|h 

15 TR A, B, r fiigrj. iara 6 0. intl i & gj^et za A, 
B, r fiBQTj, 6 & aga imb bfiavviioiv aQi&fiav (lergi]^ 
otrat rolg A, B, r (iipEatv. rotg Si A, B, F itfffsaiit 
o{u6w(ioi Kfft&fioi fiatv oi ^, E, Z' o @ aQtt viw 
rav A, E, Z fiertfeVtai. xai iartv ikdaaav %ov H' 

20 Offfp ietlv ddijvatov. ovx aga ierat rtg row if Hds- 
aav aQt^fios, os *!*' ^^ -^j B, V fiierj' onsQ ^Sec Seti/u. 

1. laayus] om. p. 3. fia' BV, P m. rec,- fi|3 p. «. 

ietio iB So9(vta fiiqi]] Bupra m. 1 p.^ 8. fffruffo:*'] -aar 

Bupra V. ytJp] om, BVpqj. 9. xal t(lTiip9(o wno rav jd,K,Z] 
mK. qs. vTio BVpy. 10. Poat o H add. Theon: insl {htll 
oJv V9, Koi inei P m. rec.) 6 M vao xav J, E, Z fifieiftai 
(BVpcp, P m. rec). 11. «H Pp, om. BV91. 12. inC\ 
isTiv PB, om. p. la] om. P. Tj supra ra. 1 V. 14. 

Poet (»1; add. Tkeon: H ilttxitsroe aiv t%ti ta A, B, r iti^ 
{BVp93, cl yuff liTj H lldiustog av mg. gj). Savat] lazio Pp. 
its] snpra m. 2 V. 16. fieB»)] om. P. 19. ilatxiav P. ai. 
Ante oet&fioK eta*. og V. In fiae.- Eoxltrtou ffioiieito»' f PB. 



ELEMENTOEUM LIBEK \1l. 269 

permutatim ^ unitas Qumeruiii B et T numerum ji 
aequaliter metitur [prop. SVj. ergo F numerum A 
metitur; quod erat demonstrandum. 

XXXTX. 

Numerum inuenire minimum, qui datas partes lia- 
beat. 

_ „ Sint datae partes A, B, F. 

h — -I I- 1 t- 1 oportet igitur numerum inuenire 

I ^ , , ^ I minimum, qoi partes J, B, F 

I iZ „ habeat. 

I 1 A partibua enim A, B, F de- 

' " ~ "^ " ^**^ nominati sint uumeri ^, E, Z, 
et snmatur*} numerus H, quem ^t, E, Z minimum me- 
tiantur. H igitur partes babet a numeris z/, E, Z 
denominatas [prop. XXXVII]. uerum n J, E, Z de- 
nominatae partes sunt A, B, r. itaque H partes A, 
B, r babet. iam dico, eum etiam minimum esse. 
nam si minus, erit numerus aliquis numero // miuor, 
qui partea A, B, F habeat. sit &. quouiam & par- 
tes j4, B, r babet, numerum & metieatur numeri 
s partibuB A, B, F denominati [prop, XXXVIII]. 
uerum a partibus A, B, F deuominati suut numeri 
^, E, Z. itaque uumerum uumeri i/, E, Z metiun- 
tur. et minor est numero H; quod fieri nou poteat. 
ergo non erit numerus numero H minor, qui partes 
ji, B, r habeat; quod erat demonstrandum. 

1) Itaque EnclideB hio qaoqae prop. S6 de tribuB tftntum 
numeriH denonBtcatam tacite ad quamlibet numerornm multitu- 
dmem trauetulit, Hicutt aupra iu prop. 33 eodem modo prop. 3 
tacite dilatauit (u. p. S66 not.). 






/ 

CC . 



'Eav m6iv o6oiSri%otovv aQLd^fiol s^ijg dvi' 
Xoyov^ ot S\ &XQ01 avtmv TtQStov TtQog dXXi^- 
Xovg m6Lv, iXdxt6toi €l6i tmv roi/ avrov Ao- 
6 yov i%6vt(ov avtotg. 

"E^tm^av o^o6ovovv aQi^yLol i^f^g dvdXoyov ol 
A^ By r*, z/, oC Si axQOc avtmv ot A^ A^ itQ&xoi itQog 
dXXi^Xovg i6t(o6av' Xiya, otc oC A, 5, F, ^ iXd%i,6tol 
€l6i t(3v ti)V avtbv Xoyov i%6vt(ov avtotg, 

10 El yaQ fti}, i6t(o6av iXdttoveg tmv Ay B^ F^ A 
oC Ey Zy Hy ® iv t^ avtm X6y(p ovteg avtotg. xal 
iytel ot Aj Bj r, A iv tm avt^ X6yc) €l6l totg JS, Z, 
jFZ, 0, xai i6tLv t6ov ro TrA^^og \tmv A^ B, P^ ^ tS 
7tXri%^eL \tmv E, Z, H^ 0], Sl' t6ov aQa i6tlv mg o A 

16 ^Qog tov A^oE JtQog tov ®, ot S\ A^ jd TtQcototj 
ot Sh TtQmtov xal iXd%L6t0Lj ot Sl iXd%L6toL aQL^- 
[lol n€tQov6L tovg tov avtov Xoyov i%ovtag ledxLq 
t€ net^mv tbv fiet^ova xal 6 iXd66mv tbv iXd66ova^ 
tovte6tLV o te fjyoviievog tbv fiyovnevov xal 6 iito- 

20 nevog tbv eito^evov, [letQet aQa 6 A tbv E o luC- 
^mv tbv iXd66ova' OTteQ i6tlv dSvvatov. ovx aQU 



EvyLXsCSov axoi%Bioiv J : i? V. Post titulum in textu scholi- 
um ad VII, 39 habent Vpqp; u. app. 4. coffiv] om. Vqt». 
bIoiv PB. 9. daiv B. 11. W] postea insert. V. 12. 

<^] postea insert. V. Blaiv B. 13. %ai hxiv — 14: 0] mg. 
m. 2 V. 13. Tcov A, B, T, z/] om. P. 14. twv E, Z, H, 0] 



vm. 

L 

Si quotlibet nmneri deinceps proportionales sunt, 
et extremi eorom inter se primi sunt, minimi sunt 
eorumy qni eandem rationem habent 

Sint quotlibet numeri inter se proportionales dein- 
ceps j4y B, r, jdj et eorum extremi A, z/ inter se 
primi sint. dico, numeros A, B, F, ^ minimos esse 
eorum, qui eandem rationem habeani 

A\ 1 I lE 

B I 1 I 1 Z 

T\ 1 1 1 K 

A\ 1 I 10 

Nam si minus^ numeri Ey Z, H^ S numeris ^, B, 
r, A minores sint eandem rationem habentes. et quo- 
niam A^ B, F^ A et E^ Z, If, ^ in eadem ratione sunt^ 
et multitudo multitudini aequalis est, ex aequo erit 
[VII, 14] ^ : ^ = jB : ®. uerum A^ A primi sunt, 
primi autem etiam minimi sunt [Vn, 21], minimi 
autem numeri eos, qui eandem rationem habent, aequa- 
liter metiuntur, maior maiorem et minor minorem 
[Vll, 20], h. e. praecedens praecedentem et sequens 
sequentem. itaque A numerum E metitur, maior 



om. P. 18. re ^tiloiv — 19: tovt^wv] P; om. Theon 
(BVg>). 21. a8vva.xov\ atonov Vqp. 



272 ETOIXEiaN ^■. 

oC E, Z, H, @ ilaeeoves ovce; ^i»v A, B, F, ^ iv 
xa avTia loya dtslv avtoZq. ot j4, B, F, ^ apa 
ildxtatoi dai tmv tov avtbv loyov exoi/tov avrois' 
omQ idei SsC^ai. 

'Api&ftovg svQstv £%ijs avaXoyov ^Aax^ffTOug, 
oaovg tiv i:iitai,ri rtg, iv ta do^&fvrt Xoy^. 

'Eata 6 do&tls loyos iv ikaxietois aQt9(tots o 
roij A npos xov B' Sti d^ agi&fiovg evpeiv i^^g 
10 avaloyov dXaxietovs, oOovs av tis inita^jj, iv tp 
rot! A ngos t:bv B koyat. 

'Ent.Ttrdx^<>}attv Sij TteaaQBs, xal 6 A iavibv 
nollanXaaideas rov F aoisCta, tbv 6i B xoXXaxltt- 
aideas tbv ^ zotBites, xal hi o B iavtbv noAAa- 
15 nXaeidaas tbv E noisizGi, xal Ixi o -A tovg f, ^, E 
jioAAK7rA«8(Kff«g rot/g Z, H, & noieha), 6 3h B xbv 
E noXXajtlaaidaas rbv K aoieiTm. 

Kal iittl 6 A iavtbv ftiv aoXXaaXaaidaag tbv 
r maodjxtv, xbv fli B TtokXanXaaidaas xbv ^ m- 
20 noiijxtv, ^atti' «pa d>s o A repog tov B, [ovrtos] 
b r W(>og Tov j^. naXtv, ixsl o fiev A xbv B noA- 
XaTtXaaidaaq xbv z? jtmoitjxtv, b Si B iavxbv TtolXa- 
aXaatdaag xbv E nEaoiijXEV, ixdxtQos aga 
tbv B aoXXaaXaaidaas ixdtsQov xmv ^, E XEJto£t)i 



3. ilaiv P. av-iols} om. V^. 7. ng ^jrtrogj; P. 
j|)js] Bupra m. 2 V, om. q:. 10. ^jKwiij ns \ ip. la. 
rc<ieaeeg] ^ P et poat ras. 1 litt. B. 13. ibv di B — 14: 

Jioiaiito] oni. (p. 18. ftc*] om. V(p. 19, jttnoiTjiiiv] (prina) 
nf^oCiiv.s Vq). 20. Ante {«tiv add. Tbeoa: aetOiios Sn o A 
iSvo Tovg J, B noHajilaaidaas tavs F, ii nejcoiTjttfv (BV9). 
t6v] inaert. ip, oJroie] oni. P. 21. /ifu] P, om. BV9, 

24. Tiu»'] tov P. 



1 



ELEMENTORUM LDBER VUI. 273 

minorem; quod fieri non potest. itaque E, Z, H, & 
eandem rationem non habent ac A, B, r, jd, quibus 
minores sunt. ergo A, B, F, z/ minimi sunt eorum, 
qui eandem rationem habent; quod erat demonstrandum. 

n. 

Numeros inuenire minimos deinceps proportionales 
in data proportione, quotcunque propositum erit. 

Sit data proportio in numeris minimis^) A:B, 
oportet igitur numeros inuenire minimos deinceps pro- 
portionales in proportione A i B^ quotcunque propo- 
situm erit. — propositum sit, ut quattuor inuenia- 
mus, et sit AXA = r, AxB = ^, BxB = E, 
Axr=Z, AXA = H, AXE = ®, BxE = K. 

— \A I ir 

\B I \d 



E 

H 



9 

K 

1 



et quoniam ^X^ = ret Ax B = A, erit 

A:B = r:A [VII, 17]. 

rursus quoniam AxB = A etBxB = E, uterque 

A, B numerum B multiplicans utrumque A, E effecit. 



1) Si proportio data minimis numeris proposita non est, 
per. YII, 83 minimos inueniemus eorum, qui eandem ratio- 
nem habent. 



Enolides, edd. Ueiberg et Menge. II. 18 




274 STOrXEIiJN Tj". 

iaiiv apa mg o A jipos zov B, ovrias ' 
lov E. akk lag o A agog zov S, o T Jtpog loi' 
^" xal d>g «pa 6 f srpos tow ^, 6 ^ jcpoff loi' £. 
xai ixfl o A Totig f, A JtollanXaeiaeas rovg Z, ^ 
6 mnairjxtv^ letiv uqk ag o F npog rov ^, [oiJraj] 
6 Z JTpog rov H. cos di 6 r" 3rp6s tov ^, oiJtMg 
fiv o A npoq toi' B' xai ibs «pa 6 ^ n:pos lov B, 
6 Z ffpog Tov H. iciiliv, iiCBl o A rovg A, K «oA- 
Xan}.aeiK6ag zovg H, @ iiexoItixbv , lotiv UQa ms *> 

10 A ngos rbv E, o H jrpog Tor &. alX' tas o A 
jcpoff row E, 6 A Jtpog t6w B. xal tbg &Qa 6 A 
jrpoff Toi' B, ovtajg 6 if XQos roi' 0. xai ijiti of 
.^, B Tov ^ XQ^lazXaetaeavTBS Tois ©, JC ^sjioi-^xa- 
6iv, iaxtv aga ms A Jtpog tou B, owtojs 6 ® 

15 Tpog tov -ff. «ii' dis o A iipos tof B, oCtcjs o 
TJ Z Jipog tov H xal 6 H jrpos tov ©. J^al rag opa 
o Z Jtpo;; Tov Jif, ovtws o ts H npog tbv & xat 
6 jrpos Tov K- 01 r, ^, £ apa xal oC Z, i/, 
0, X RvaAoyov ftov ^v ta toii .^ ffpos tov B koya. 

20 AiV*^ '^'it ^'^' "'^^ iXaxietoi. inal ydp ot ^, B ^A«- 
Xietoi sCei tiav tbv avtbv -loyov ixovtcov avzoZs, 
of ds iXKXtexoi tmv tbv avtiv Adyow ixovtfav Jipd- 
Tot Jipog aAAi^^ovg sleiv, oi ^, B Rpa JEpfotoi. npos 
aXXtjXovg sieiv. xal sxdrsQog ^lv riov A, B iavrov 

2a noXXaxXaeideag ixdtsgov ttov V, E zenoirjXSVf sxa- 
TSQOV dh trav r, E aoXXttTtXaeideag sxdtspov xav 
Z, K XBJtaiijxsv ot r, E apa xal ol Z, K Jipcnrot 
3ip6s dXX^Xovg sleiv. iav S% laeiv oxoeoiovv dgi^- 
ftol i|^s dvdXoyav, ot S^ dxQOi avttoi' jtpratot ffpoff 



I V , OVTOie 



ELEMENTORUM LIBER Vm. 275 

itaque j4 : B = J : E [VII, 18]. uerum u4:B = r:^. 
quare etiam F: ^ = ^ : E, et quoniam Jl X r= Z 
ei j4xJ = H, erit F : ^J = Z : H [VII, 17]. uerum 
erat F : /d = A : B. quare etiam A:B = Z:H. rur- 
sus quoniam AX /d = H ei AXE = 0, erit [VII, 
17] ^ : E = H: &. uerum /d:E = A:B. quare etiam 
A : B = H : ®. et quoniam 

AXE = & et BxE = Ky 
erit [Vn, 18] A:B = &:K. uerum 

A:B = Z:H=H:&. 
quare etiam Z:if=iJ:® = ®:iL. itaque F, z/, E 
et Z, iJ, @, iC proportionales sunt in proportione A : B. 
iam dico, eos etiam minimos esse. nam quoniam A^ 
B minimi sunt eorum, qui eandem rationem habent, 
minimi autem eorum qui eandem rationem habent, inter 
se primi sunt [VII, 22], A^ B inter se primi sunt. et 
uterque A^ B se ipsum multiplicans utrumque F, E effecit, 
utrumque autem JT, E multiplicans utrumque Z, K effecit. 
itaque F, E ei Z^ K inter se primi sunt [VII, 27].^) 
sin quotlibet numeri deinceps proportionales sunt, et 
extremi eorum inter se primi sunt, minimi sunt eorum. 



1) H. e. r et £ primi sant inter ee et item Z et K, nu- 
meros F^ E, J corollarii causa per totam propositionem 
respicit. 

xal 6 J (p. E] e corr. V. 4. tovg] corr. ex xov V. Tovg] 
corr. ex tov V. 6. ovroog] om. P. 8. H] seq. ras. 1 litt. V. 
10. 6 H] ovtmg 6 if qp^ et m. 2 V. dU' dag] mg di P. 12. 
ovtmg Ticci P. 14. ovtmg] om. BV9. 15. all* ] ideix^^ 
81 %al Theon (BV9). 17. t^] om. P. 19. Zoyflol supra 
m. 2 B. 21. daiv P. avtorg — 22: ixovtmv] om. P. 22. 
Post ixovtoov add. avtOLg V9, et supra m. 2 B. 24. Bia£Y(p. 
27. K] (alt.) H 9. 29. di] om. tp. 

18* 



I 



276 STOIXEISiN ri'. 

dkkrjXovg m6cvy ikdxt6tot Bi6t xAv tov avtov Xoyov 
i%6vtiov avtotg. ol F^ /d ^ E aga xal ot Z, ii, ®, K 
ika%t6tot 6l6i t&v tbv avtov koyov i%6vrGiv totg 
A^ B' OTtSQ idst dst^ai. 

5 n6Qt6(ia, 

'Ex Sri tovtov g^avsQov^ ott idv tQstg ccQt^fiol 
i^ijg dvdkoyov ikd%L6toi (o6t tmv tov avtov k6yov 
i%6vt(ov avtotg^ ot axQOt avtmv tstQdycovot al6tv^ 
idv 8\ ti66aQsg, xvfiot, 

10 y'. 

^Edv (o6tv o7Co6otovv aQtd^fiol «l^? dvdko- 
yov ikd%i6tot tmv tbv avtbv k6yov i%6vt(ov 
avtotg^ ot axQOt avtAv TtQcitot n:Qbg dkkTjkovg 
si6tv. 

15 "E6t(o6av bn:o6otovv aQtd^^ol s^ijg dvdkoyov ikd- 
%L6toL tmv tbv avtbv koyov i%6vt(ov avtotg ot A^ 
B, jT, ^' ksycDj ott ot axQOt avtciv oC A^ ^ JtQcotOL 
TtQbg dkkrikovg si6Lv. 

Eikricpd^Gi^av yaQ 8vo fihv dQL&^ol ikd%t6tot iv 

20 tcp tciv A^ B^ F, /d k6y(p ot E, Z, tQstg Sh ot ff, 0, 
K, xal s^^g ivl Jtksiovg^ smg tb kafipavo^svov Tckij- 
^•og t6ov yivr^tat tS Ttki^d^SL tSv A^ B^ jT, ^. siki^- 
(pd^m^av xai s6tc36av o[ A, M, iV, S. 



1. siaiv PB. 2. K] corr. ex T m. 2 V. 6. noQiaiia] 
mg. m. 2 V, om. cp. 6. idv'] ccv seq. ras. 2 litt. P. 7. 

aaiv iXd%iatoi Vqp. mciv B. Xoyov] mg. cp. 9. 8s] supra 
m. 2 V. -cBOGaqBq] d B. 17. F] postea insert. m. 1 V. 
20. ot H] corr. ex ot m. 2 B. 21. K] in ras. P. %aC] 

supra add. at m. 1 P; v.al dei B. ccog ov Theon (BVqp), 
ewff av August. 23. iGttaaav] -v e corr. m. rec. P. 



ELEMENTORUM LIBER Vm. 277 

qui eandem rationem habent [prop. I]. ergo F, z/, E 
et Z, ii, @, K minimi sunt eorum, qui eandem ratio- 
nem habent ac ^, £; quod erat demonstrandum. 

Corollarium.. 

Hinc manifestum est, si tres numeri deinceps pro- 
portionales minimi sint eorum, qui eandem rationem 
habeant, extremos eorum quadratos esse, sin quattuor, 
cubos.^) 

m. 

Si quotlibet numeri deinceps proportionales sunt 
minimi eorum, qui eandem rationem habent, extremi 
eorum inter se primi sunt. 

Sint quotlibet numeri deinceps proportionales j4, 
J5, r, ^ minimi eorum, qui eandem rationem habent. 
dico, extremos eorum J, z/ inter se primos esse. 

^ „ sumantur enim duo 

numeri minimi in pro- 

„ portione numerorum j4, 

B, r, J [VII, 33] E, 

Z, tres autem H, @, K 

^ et deinceps uno plures 

[prop.II], donec multitu- 

'^ do sumpta aequalis fiat 

multitudini numerorum ^, B, F, z/. sumantur et sint 

^, M, N, S. et quoniam E, Z minimi sunt eorum, 

qui eandem rationem habent, inter se primi sunt 



E 



\H 10 

\K 



\A \M 



1) Nam >i : B = r : z/ = z/ : E et r = >i», E =» £*• 
praeterea AxB ^ Z : H^H:©^e:K et Z^Ax r^Ah 
K^BxE=-B\ 



278 STOIXEISiN fi'. 

Kal iTtel oC E^ Z ika%i6toC €l6v tmv xov avtbv 
Xoyov i%6vr(ov avtolg^ TCQ&toi yCQog akX^qXovg st^lv, 
xal insl ixdteQog tmv E^ Z iavtov (ihv TtoXXaxXa- 
6id6ag ixdtBQOv tmv iJ, K 7tsJto(riX£v ^ ixdreQOV di 
6 t(j5v iJ, K nolXa7tka6id6ag ixdtsQOV tmv A, S ^s- 
TtoCriTceVy koI ot f/, K aQa Ttal oC A^ S JtQciroi^ XQog 
dklijXovg el6Cv, xal iitel oC A^ B^ F^ jd iXdxt6toC 
eC6L tSv tov avtbv Xoyov ixovtcov avtotg, el6l di 
xal oC Aj M^ N^ S iXd%i6tov iv tS avtS Xoym ovteg 

XO tolg Ay Bj r, ^j xaC i6tiv t6ov tb nXfi%^og t(5v A^ 
Bj Fj /J tm 7tXri%^ei tmv A^ M, iV, ^, exaatog aQa 
tmv Aj Bj r, z/ exd6t(p tmv A^ M, N, S t6og i6tCv' 
taog aQa i6tlv 6 filv A t^ Ay b 8% /d t^ S. xaC 
eL6Lv oC Aj S TtQmtoL TtQbg dXXi^Xovg, xal oC A^ A 

15 aQa ytQoitOL TtQbg dXXriXovg eC6Cv' ojteQ iSeL det^aL. 

8\ 
Aoycav Sod^evtcov bito^cavovv iv iXa%C6tOLg 
aQLd^fiotg aQLd^fiovg evQstv el^ff dvdXoyov iXa- 
%C6tovg iv totg Sod^et^L XoyoLg, 
20 ''E6tco6av 01 Sod-evteg Xoyov iv iXa%C6roLg dQLd"- 
^otg o te tov A TtQbg tbv B xal b tov F TtQog 
rbv A xal etL b tov E TtQbg tbv Z* Set Srj aQtd'' 
^ovg evQstv «l^g dvdXoyov iXa%C6tovg iv te rp rou 
A TtQbg tbv B Xoyc) xal iv rc5 toi) F JtQbg tbv A 
26 5«al itL iv t(j5 tov E TtQbg tbv Z. 

ECXi^^pd^ca yccQ b vitb rc3i/ 5, F iXd%L6rog ^erQov- 
Hsvog aQLd^fibg b H, xal b^dxLg ^sv 6 B tbv H 

1. xal insC — 3 : eccvtov ^iv] ot ccqcc cchqoi avtav ot A, S 
nQmtOL nQog ciXXi^Xovg slaCv. insl yccQ ot E, Z nQcitoL EtidtSQog 
Ss Kvtmv sccvtov Theon (BVqp). 1. sleiv P. 4. A] eras. V. 
5. tmv A^ tov A P. 6. Ha^J om. BVqp. xal ot A^ ISI — 7: 



^ ELEMENTORUM LIBER VEI. 279 

' [VII, 22]. et quoniam E x E '^ H, Zx Z = K 
[prop. n coroll.] et ExH= A, ZxK=S [id.], 
et H, K et A, S inter se primi sunt [VII, 27], et 
quoniam A, B, F, d miiiimi sunt eorum, qui eandem 
rationem habent, et etiam A, M, N, S minimi suut 
ia eadem ratione ac A, B, F, /i, et multitudo uume- 
rorum A, B, F, ^ raultitudiui uumerorum A, M, N, S 
aequalia eet, einguli A, B, f, ^ siugulia A, M, N, S 
aequales sunt. itaque A = A, zJ = S. et A, S iuter 
86 primi sunt. ergo etiam A, ^ iuter se primi Bunt; 
quod erat demoustraiiduu]. 

IV. 

^ Datis quotlibet rationibus in numeris minimis au- 
meros inuenire minimoa deinceps proportionaleB ') in 
rationibus datia. 

Siut datae rationes in uumeris minimis A : B, 
■X'--A, E;Z. oportet igitur numeros minimoa inuenire 
4«inceps proportionales in rationibus 
A:B,r:^,E'.Z. 
sumatur euim, quem minimum metiuntur B, F, 
lumerus H [VII, 34]. et quoties B numerum H me- 



I 1) Uerba e|^s dvaXoyov lioc loco proprio seneu nsurpata 

«ou «aoi; neque enim ratioues inteir ae aequales sunt. Bigoiiicat 
,Eiiclidea, termiEuni sequentem prioriB rationiB praocedentem 
™^— " poBterioris. habet idem Campanus. 

elnlv^ sQ^thi xol of A, S Theon (BV<p). 7. mI Imi — 8: 
tlai'} mg. m. 1 P. 7. ^] om. B. 8. tlvt] elaii' P; loai Vq,. 
B. haparoi] om. V<p. 14. tleLv} P; inii Theon (BV<p). 

_ FoBt aUiiXovs add. Theon: ile^v, hog 3i o fi.iv A tm A 6 Sl 

m^S ra J (BV<p). 18, aviloyov] 'P; V mg. m. 1, del. m. reo.; 

mMta. BcB. 19. So9eC<nv B. -21. to»] corr, es lo V. *sa. 

Kt^] seq. raB. 2 litt V. 23. dvaloyov] om. BVqi. 




278 STOIXEiaN fi'. 

Kal ijtsl ot E^ Z ika%v6tol slct x&v rov avtw 
koyov ixovtmv avtotg^ TCQmtoi iCQog akkriXovg bMv. j. 
xal iTCsl iocatsQog tmv E^ Z iavtov fihv TCoXXaxl»' : 
6ia6ag BTcatBQOV xAv H^ K TtSTCoCrixev , ixdrsQOV A^ 
6 tojv H^ K 7Colka7cka6ia6ag ixdtSQOV x&v Aj S 
TCoCrjxsv, xal ot H, K ccQa xal oC A, S TCQmxoc 
dkkijkovg sleiv. xal iTCsl ot A^ B^ F^ ^ ika%i 
si6L xmv xov avtov koyov i^ovxc^v avxotg, sM 
xal 01 Aj M^ N, S ikd%L6toi iv x& avxip koym 

10 Tor^ Ay B, r, Aj xaC i6tiv t6ov xo nkijd^og xmv 4t 
Bj Fj A rc5 nki^d^sL x6v A^ M, iV, S, sxa6xog ^ 
xciv Aj Bj r, A sxd6xfp xmv A^ M, iV, S ^^og i6%k 
taog aQa i6tlv 6 filv A tm A, 6 Sh A tp S, m 
SL6LV ot Aj S JCQoitoL TCQog dkki^kovg. xal oC A^ * 

15 aQa TCQmtoi iCQog dkkrikovg si6Cv otceq iSsL Ssi^/a$» 

S\ 
AoycDV Sod^svxcDV 67C06C3VOVV iv ika%C6%( 
aQLd^fiotg aQLd^^ovg svQstv «l^g dvdkoyov i/ 
%C6xovg iv xotg Sod^st^L koyoLg, 
20 *'E6tG)6av OL Sod-svtsg koyoL sv ika%C6tOLg dc 
^otg ts tov A TCQog rov B xal 6 tov F tt 
rov A xal sxl 6 tov E iCQog tov Z* Sst Sri &• 
^ovg svQStv s^rjg dvdkoyov ika%C6tovg Iv ts x^ 
A TCQog tbv B k6yc3 xal iv xS toi5 F ^Qog ro. 
25 xal IxL iv xc5 xov E TtQog xov Z. 

ElkT^fpd^ci) yaQ 6 vTCo xciv J5, JT ikd%L6xog (iS' 
Hsvog dQLd^^og 6 H, xal b^dxLg ^hv b B xb 

1. xal insi — 3: savtov fiiv'] ot aga utiqoi avtmv o 
TtQ&toi TZQog dXXriXovg bIgCv. insl yccQ ot E, Z nQmtoi iv. 
ds uvtdiv savtov Theon (BVqp). 1. slciv P. 4. A] e" 
5. tmv A'] tov A P. 6. Hat] om. BVg?. xal ot A, !r 



^ 



X 



... * . I .' 1 1 1 f f • • f 

.'I ■ f • • f f I J I . •/ 




J^ ;..tittt»iiit / / • • • • 




I • 



I •• 



.* « • |-H|il :« • f/ •• • • • • 

-^ .'.f/.l !'•»••••••• ••** 



I #••• 



I t -m t * 



\ -• 



r. 



I • ■ 

'. ., ... N 



280 ETOIXEias ij'. 

(lEtfjit, zoOavxaxig xal o A xov pLBZQslxfa ^oaca^ 
6\ b r Tov H futgst, roffavzaxis ^i o ^ xhv K 
fittQsiTm. Si E tov K ^rot fietQft 7} ov fttxQtL 
fiSTpiita XQotigov. xal oaaxig 6 E tuv K (tstgct, 
6 ToeavzKxiq xal 6 Z rov A (LEtgsita. xul intl led- 
Mg o A Tov @ fiftght xal b B tov S, Istiv aga 
d>g A ZQOS tov B, ovidis 6 © ngog tov H. 
Ta avta d^ xal mg 6 F xgog TOi' ^, ovtcog 6 
jtgbg xbv K, xal Sti as b E agbg tbv Z, ovtag 

10 K TCpbs tbv A' ot ©, i/, K, A aga f|^s avdi.oyt 
sloiv iv t£ rm tov A ngbs tbv B xal iv ra tow 
Kpoj Toi' A xal itL iv rra rov E ngbg tbv Z koy^ 
kiyto 8-q, oti xal iXdxiSTot. sl yag fiij siaiv ot 
H, K, A (£^5 dvdkoyov ikaxiOtot iv ts Totg to« 

15 Jigbs tbv B xal rov F irpog lov ^ xal iv xd 
E ngbs tbv Z Xoyotg, eaTiaiSav oC N, S, M, O. 
iTCsi ioTiv <o; o A jrpog tbv B, oitGJs o N spi 
Tov S, 01 8\ A, B ikdxtStoi, of d% ikdxiotoi (m- 
rpovfft Towg Tov KUTov koyov ix^vtag iedxtg o « 

80 iisi^av tbv ftf^^ovK xal b iXdeecov tbv iideaovaf 
Tovtietiv o T£ Tiyoviisvog tbv JiyovfiEvov xal 6 
ftfvo? xbv sno^svov, b B aga xbv S ^szqsT. 

1. #] eras. V. 2. «oc] om. Vqi. 9. ht tos] i 
m. rec. P. 10. 0, H] e corr, poet raa. 2 litt. V; H, © B. 
a*oloyoi'] P; om, BVqj. 11. «1 om. V^p. 13. &} eraa. V. 
©, H] H, B. 14. avdloyov^ P; mg. m. 1 V, deT. m. roc; 
om. B^. «1 om. BV?;. 15. ko/] «nl ij- rii P. iv irro] f« 
im B, ixt Iv i^ Vqi. 16. Poat loyois adcl. V^pr ftovrtf^ nwe 
imv H, @, K, A i^ijt (mg. V) iluBeoves dgi&fiol i'v rs rott 
loi' ^ 7(Qee 'cov B xal lov P ngos Tav ^ «<el fri (aapi& V) 
Eou E xQos rof Z loyofc; ideca B mg. m. 2 om. ci-qg et li 
17. cos] eupra m. 2 V. N] H tp. _ 18. oC *i ^lap 

om.T. iietQOveiv Y f. BO. ^iiirtDi»i rot ilatxmia "Vip. 
tb] om. P. 22. aeo!] ?« (p. 



i 



] 



ELEMENTORUM LIBER Vin. 281 

titur^ toties etiam A numerum S metiatur^ quoties 
autem F numerum H metitur^ toties etiam /d nume- 
rum K metiatur. E igitur^) numerum K aut metitur 

A\ 1 B\ 1 

r\ 1 z/i 1 

E\ 1 Zi 1 



1 \N H 

1 1 



-I 



e 

I : 1 



M 1 \K 

,0 I :; 1 



aut non metitur. prius metiatur. et quoties E nume- 
rum K metitur^ toties etiam Z numerum A metiatur. 
et quoniam A numerum ® et B numerum H aequa- 
Uter metitur, eniA:B = @:H [VII def. 20. VII, 13]. 
eadem de causa erit etiam F: A = H : K et prae- 
terea E : Z = K : A, itaque @, Hy K, A deinceps pro- 
portionales sunt in rationibus A : By F : A, E : Z, 
iam dico, eos etiam minimos esse. nam si 0, Hy K, 
A non sunt minimi deinceps proportionles in rationi- 
bus A:B^ F: Ay E:Z, minimi sint iV, Sy M, O. et 
quoniam est -<^ : JB = iV : ^, et A^ B minimi sunt, 
minimi autem eos, qui eandem rationem babent, aequa- 
liter metiuntur, maior maiorem et minor minorem, 
h. e. praecedens praecedentem et sequens sequentem 
[VII, 20], B numerus numerum S metitur. eadem 



1) Uidetur enim pro 8i lin. 3 scribendum esse d^; cfr. 
p. 194, 23. 262, 11. 



282 STOIXEIiiN Tj". 

T« avta di} xal 6 F zov S ftfTpft' of B, 

5 [isTpovaiV xal ikaxi-ozo^ aga vjto xmv B. V 
lUTQOViiEvos Tov Si (iiZQ^asi. ^ARj;nlTOff dh vjio rmv 
B, r itiTQSitai, H- Q H aQa roi- Sl (isxget 6 fitl- 

5 ^atv tov ikaG6ova' oitip ieTiv advvaTov. ovx a^a 

iaovTuC Ti.v€s ^w* &' a^ -K, A ildoeovie aQi&fiol 

i^ijg Iv T£ TM Tov A TtQos tov B «al tm roti f 

jipog Tov ^ xal iti. T(5 tou B repos tov Z Adyro. 

Mi] (itTQBtTco 6^ 6 E Tov K. xal eil^tp&ai imo 

10 Toiii E, K ikdxiSToq {liTgovfitvog dQi&jios 6 M. 
xal oodxts filv o K row M ftiXQit, xoOavxtkiuq xaX 
ixdxEQOs tmv ®, H ixdxeQov xmv N, ^ (isxgiixe}, 
oadxig Ss 6 E tov M (isrpii:, xoOavxdxis xal 6 Z 
Tov O iifXQtirco. inel tsdxig o & tov N iisxQtt xal 

16 6 if Tov S, laxiv d^a rag o © jrpog tov H, owTCJg 

6 N ffpos Tov S- coff d^ 6 ® sipog xov H, ovxm 
6 A nQog Toi' B' xal ag dga 6 A ngbs xhv B, ov- 
xas o N irpos tov S- ^i-a xa avta d^ xal ros 6 T 
jtpog Tov ^, ovTcag 6 S Jipos xbv M. itdktv, iail 

so iadxis " E '^bv M (iBTQst xal b Z tov O, iaxi-v ttQU 
mg b E srpog roi' Z, outos 6 M nQog xov O' oC N, 
!:, M, O dgtt i^ijs dvdi-oyov datv iv tots tov xa A 
wpog xbv B xal tov F Jipog tov A xal ixt xov _£ 
Kpog Toi' Z koyaig. kiyot S-q, Zxt xal ikdxiOtoi 

2. fi,szeovei Y q>, vno] 
a] del. m. 2 B, mg. iteTfovfitj 
(Wieei. e. 0, H] H, & Bip et 



1. B, r]^r,^B BVip. ^ 



1 H i 



in tae. V. 7. Post eg^B in B^inBert. 
om. P. 8. iJ] ^ loyoj Vqj. loyco] om. Vqj. ' II. fiit] 
m. 2 V. M] fijj ip.' 12. 0, M] corr, oi H. V; 

Jf, FBtp. 13. M] fiTJ ^. 14. iTtii] Kol Ine^ V m. 2, q>. 
20. ftrii»' oeo — 21: i6* 0] mg. ip. 22. ovaioyop] om. 

BVip. rou] iw»' P. rf] om. Vip. 23. fri] om BVgi. 



i 



ELEMENTORUM LIBER Vm. 283 

de causa etiam F numeruiu S metitur. itaque B^ F 
numerum S metiuntur. quare etiam^ quem minimum 
metiuntur B, JT, numerum IS metitur [VII, 35]. mini- 
mum autem B, F metiuntur numerum H. itaque H 
numerum S metitur, maior minorem; quod fieri non 
potesi itaque nulli numeri numeris 0, if, K, A mi- 
nores deinceps in rationibus A : B, F : ^, E : Z erunt. 
ne metiatur igitur E numerum K. et sumatur, 
quem minimum metiuntur E, K, niimerus M [VII, 34]. 

A\ 1 n ! E\ 1 



B\ 1 J\ 1 I- 



H\ 1 I 1(9 

I \K I ilT 

M\ 1 I \P 

^i . 2\ 1 

N\ : . I 1 T 

O I 1 



et quoties K numerum M metitur, toties uterque 0, H 
utrumque N, S metiatur, quoties autem E numerum 
M metitur^ toties etiam Z numerum O metiatur. quo- 
niam ® numerum N et H numerum S aequaliter me- 
titur, erit &:H^N: S-[VII def. 20. VII, 13]. uerum 
® :H= A: B. quare etiam A : B = N: IS. eadem 
de causa etiam F : z/ = S : M. rursus quoniam E 
numerum M et Z numerum O aequaliter metitur, erit 
E:Z = M:0 [Vll def. 20. VII, 13]. itaque N, S, 
M, O deinceps proportionales sunt in rationibus 

A:B, r:Jy E:Z. 

24. iXd%iazoi staiv Y(p. Dein add. BVqp: sl yap fi^ atciv 
iXoixiazoi (om. B) ot N, ISl, M, O i^^g {iXaxiavoi add. B). 



284 ETOIXELQN »)'. 

«ore A B, r ^, E Z loyoig. c( yiiQ fiij, iaomei 

TI.VES zmv N, S, M, O ikdeeovts agi&ftol e^^s dvd- 
Xoyov iv Tofg j4 B, f ^, E Z loyois- ESraattv ot 
n, P, £, T. xal insC iettv mg 6 n Jtpo? t6v P. 
B ovirag 6 A apog lov B, ol 6i A, B ilaxiaroi, oi 
Si iXaxiatot ^atQOvet tows toi' avzov koyov sxovtas 
ttvtoTs iSttKig o XE riyoviiBvog roi/ •t^yovfievov xal h 
iftanEvos ^ov ixofiEvov, 6 B «p« t6i' P (letget. dii 
ta avta 6i) xal 6 F t6v P fLbtQEt- ot B, F aga roc 

10 P ftiTQovaiv. xai 6 dS,dxtetos Sqk vith rav B. f 
fittQovfievog tov P fietQ-^eei. iXttxtGtog dl vno tmv 
B, r ^ezQOV^evos iottv 6 H' 6 H aga tov P fietQet. 
xai iariv mg o H wpog tov P, ouTtag 6 K nQog tov 
S' xttl 6 K ctQa tov E fietQeL (letQet Se xttl 6 E 

IB Tov £' ot E, K ttQu tbv U fistQoveiv. xal 6 iia- 
XmTtos ttQcc vito tmv E, K fittQovfievog rbv 2 fie- 
tQTJeH, ikdxtetos ^E vitb T(oi/ E, K fietgovfisvos 
iettv 6 M' 6 M aQa tov 2/ ftetQst b fiei^fov tbv 
iXdesova' oitEQ isrlv dSvvarov. oiix aQa eOovtai 

20 TtwEs rSv N, S, M, O ildeeoveg aQi&fiol e^^g ^vd- 
loyov iv tE rots toii A XQog tbv B xal tov P i 
rbv A xttX iri tov E itQos i:ov Z Xoyoig' ot N,} 
M, O aQtt iS,rjg dvdioyov ikdxiatoi elGiv iv totg ii 
B, r J, E Z loyoig- oiteQ ISet SEliat. 



avd- 
o m1 " 



1. ^, E, 2] om. B. ei yag fit)] Qm. BVy, 2. if] 
H rp. dvdloyov] ota. BVqj. J. m] otn. BVqj. 10. fit- 
Teovai. Vif>. 11. llaxiaios di vtcb Ttav B, F jiftgovitivos] 
6 Si iXaxiatos V9. 12. H] mutat in m. 2, anpra H 
m. 2 B. H] item B. p.itBi<ift Vip. 13. H] ati aupmB, 
15, aga] hi m. 18, £1 oorr. ex E Y- 20. ivtiXoyoti" 

ora. BVq,. 21. tov] om, B. 22. to»] om. B, 1«] ei B 
tiv] om. B. 23. dvdloyOT] om. BVqo. h] 




ELEMENTORUM LIBER Vm. 285 

iam dicOy eos etiam minimos esse in rationibus 

j4:Bj r.Jy E:Z. 
nam si minus, numeri numeris N^ S, M, O minores 
deinceps proportionales erunt in rationibus 

A:B, F:^, E:Z. 
sint 77, P, U, T. et quoniam est 12 : P — A : B, et ^, B 
minimi sunt, minimi autem eos, qui eandem rationem 
habent, aequaliter metiuntur praecedens praecedentem 
et sequens sequentem [VU, 20], B numerus numerum 
P metitur. eadem de causa etiam F numerum P me- 
titur. itaque -B, F numerum P metiuntur. quare etiam 
quem minimum metiuntur JB, F, numerum P metietur 
[Vn, 35]. quem autem minimum metiuntur B, F, est H. 
itaque H numerum P metitur. et H: P^^s K : U}) 
quare etiam K numerum U metitur [VII def. 20]. 
uerum etiam E numerum U metitur [VII, 20]. itaque 
E, K numerum £ metiuntur. quare etiam quem mi- 
nimum metiuntur E, jRl, numerum U metietur [VII, 35]. 
quem autem minimum metiuntur E, K, est M. itaque 
M numerum £ metitur, maior minorem; quod fieri 
non potest. itaque nuUi numeri numeris iV, S, M, O 
minores deinceps proportionales erunt in rationibus 
u4 : B, F: jdy E : Z. ergo N, S, Af, O minimi sunt 
deinceps proportionales in rationibus A:B,r:^,E:Z\ 
quod erat demonstrandum. 



1) Nam H: i:= F: z^ (p. 280, 8) = P: Z. tum u. VII, 13. 



286 STOIXEISiN 1?'. 






OC inCneSoi ccQtd^fiol ^Qog akXrikovg koyov 
B%ov6i xov 6vyx6Vfievov BTC tmv nkevQciv. 

**E6t(o6av inCneSoi ccQLd^fiol ol A^ jB, Tcal xov (ilv 
6 j4 TckevQal i6x(06av ol jT, ^ ccQi^fioCj xov 8e B oi 
E^ Z* Xeyc^, oxi 6 A iiQog xov B koyov e%ei xov 
^vyxeCfievov ix xAv TtkevQAv. 

Aoyov yccQ Sod^evxcav xov xe ov i%ei 6 r JtQog 
xbv E Ttal 6 A TCQog xov Z eCk^^g^d^cj^av ccqi^^oI 

10 eiijg ika%i6xoi iv xotg F Ej A Z koyoigy ol H, 0, Jf, 
S6xe elvai (hg fiiv xov F TCgog xov E^ ovxag xbv 
H JtQbg xbv ©, mg S^ xbv A itQbg xbv Z, ovxfag 
xbv ® TtQbg xbv K. xal 6 A xbv E jtokka7cka6td6ag 
xbv A jtoteCxG), 

16 Kal inel o A xbv fiev F xokka7tka6cd6ag xbv A 
TteTtoCrixeVj xbv Si E icokkaicka^id^ag xbv A TCeTCoCrjTceVj 
e6xLv aQa cog 6 F TCQbg xbv E^ ovxayg b A TCQbg xbv 
A. mg Se 6 r TtQbg xbv E^ ovxog 6 H TCQbg xbv 0' 
xal cjg ccQa b H TCQbg xbv 0, ovxcag 6 A JtQbg xbv 

20 A. jtdkiVj ijtel E xbv A 7tokka7tka6id6ag xbv A 
TteTtoCrixev j dkkd firjv xal xbv Z Ttokkaitka^id^ag xbv 
B TteTtoCrixev j e6xiv ccQa mg b A itQbg xbv Z, ovxog 
6 A TtQog xbv B. dkV mg b A JtQog xbv Z, ovxog 
6 ® JtQbg xbv K' xal cog aQa 6 ® TtQbg xbv K^ ov- 

25 xcjg 6 A jtQog xbv B. iSeC^d^ri Se xal cjg b H itQog 
xbv ®5 ovxcjg A TtQog xbv A' Sl^ l'6ov aQa i6xlv 



4. fisv'] om. P. 8. yccQ] dsC q>. 11. tov H] 6 H P. 

12. xov J] 6 J T. 13. Httl 6 J — 14: noLsCtco'] om. Theon 
(BVqp). eorum loco habent BVqp: oi aQcc H, 0^ K ngog 
dlXrjlovg l^orcrt tovg tmv nXsvQoav Xoyovg. dXX' 6 zov H TCQog 
zbv K Xoyog avyHSLzai ^x tov tov H nQog xov S xal xov zov 



ELEMENTORUM LIBER Vm. 287 

V. 

Numeri plani inter se rationem habent ex lateribus 
compositam. 

Sint plani numeri j4y B, et numeri ^4 latera sint 
r, z/, numeri B autem E, Z. dico, esse 

A:B=- r.Ex^ :Z. 
\A nam datis rationibus 



\B r:E et ^:Z^) 

sumantur numeri deinceps 



E I I z minimi in rationibus F : -E et 

iH ^ : Z [prop. IV] H, @, K, ita 

-\9 ut sit r:E^ H:& et 
-\K J:Z = S:K. 
1^ et sit jdxE — A, 



et quoniam ^ X r^= A et A X E = A, erit 
r:E = A:A [VII, 17]. uerum r:E = H:®. quare 
etiam Hi® = A : A. rursus quoniam Ex ^ = A 
[Vn, 16] et ^ X Z = By erit A:Z = A:B [VH, 17]. 
uerum zl :Z = ® : K. quare etiam S : K = A : B. 
demonstrauimus autem, esse etiam H:@ = A:A. ergo 



1) Si liae Tationes minimis numeris propositae non snnt, 
per VII, 33 minimos nnmeros inueniemus, qoi easdem ratio- 
nes habent. 



S TtQog tov K. o H aqa nQog tov K loyov ii^i xov avymC- 
fuvov i% zmv nXsvQmv. liym ovv, oxi icxlv dtg o A nQog xov 
B (in ras. B), ovxmg o H VQog xov K\ pnnctis del. V. Dein 
add. BVqp: 6 /i yuQ (B, V m. 1; xal 6 zi V m. 2; %al 6 d 
nQog (f) xov E nolXanXaaiMCag xbv A noulxa. 15. xa^] 

om. BVqp. 6 J] di q). 16. nsnoCrjns Y (p. 17. £] postea 
insert. V. 20. 6] 6 ^iv P. 22. ovxmg 6 A — 23: nQog 
xov Z] mg. (p. 



288 STOIXEKiN ri'. 

(og 6 H JtQog tov K, [ovrog'] 6 A nQog rov B, 6 dl 
H n^Qog toi/ K Xoyov i%Bi tor 6vyxs£fL6vov ix r£v 
TtkevQcov' xal 6 A &Qa ^Qog rbv B koyov l%si rov 
6vyxBiybBvov ix rciv TtXBVQmv otibq iSsi det^at, 

6 s:'. 

^Eav m6iv ono6oiovv aQid^fiol i^^g &va- 
koyov^ h 8b TtQmrog rov^dBvrsQOV iirj iistQyj 
ovdh aXXog ovSelg ovSiva (iBrQT^6BL, 

^E6rG)6av o7to6oiovv ccQtd^iiol i^ijg dvdXoyov oC 

10 A, Bj r, ^, Ey 6 d} A rbv B iiii ybBrQBCno' Xiya, 
ort ovdi aXXog ovSslg ovSiva (isrQri^Bi. 

^Ori (iBv ovv oC Ay B, JT, ^, E i^ijg akXijXovg ov 
(iBrQov6tv, g^avBQov ovds yccQ 6 A rbv B (letQBt. 
kiycD 8yi, ort ovS\ aXXog ovSslg ovSiva (ABrQi^66L. bI 

15 yaQ Swarovy (iBrQBircj 6 A rbv F. xal o6ol ei^lv 
ol A^ B^ JT, ro6ovroi Blki^g^d^co^av ikd%L6roL aQLd^ybol 
rcSi/ rbv avrbv koyov ixovrcov rotg A, B^ F oC Z, H, 
&. xal iitBl ot Z, H^ S iv ra avnp Xoycy eiel roig 
A^ J5, JT, xai i6rLv t6ov ro TtXiid^og rcov A, B^ F ra 

20 Tt^Tjd^BL rdiv Z, H^ Sy Sl^ t6ov aQa i6rlv cog 6 A 
TtQbg rbv F, oijtcog b Z TtQbg rbv &. xal iTtsi i6rLv 
(og A TtQbg rbv 5, ovrcag b Z itQbg rbv H^ ov 
(iBtQSt SboA rbv B^ ov (iBtQBt ccQa ovSh o Z rbv 



1. ovTODs] om. P. Al in ras. P. zov] om. P. 2. 
tov K] K P. rov] corr. ex z6 €p, 8. fisxQstasi qp, sed 

corr. 12. E] om. qp. ov] m. rec. P. 13. fisTffovai, 

P m. 1, Vqp; iisTQi^aovGi P m. rec. 14. si yuQ dvvatov^ (is- 
TQSLTGi 6 A Tov F] Xsyo) yocQ, OTt ov fiSTQSt o A Tov r Theon 
(BVcp). 15. Kffl oaoi] oaot yocQ Theon (BVqp). 18. sC~ 

aCv PB. 21. ZJ Z, K B. 



ELEMENTORUM LIBER Vm. 289 

ex aequo erit [VII, 14] H: K— A: B. uerum 

H:K=r:ExJ:Z.') 

ergo etiam ^: B = T: Ex J : Z-^ quod erat demon- 
strandum. 

VI. 

Si quotlibet numeri deineeps proportionales sunt, 
et primus seeundum non metitur, ne alius quidem 
ullus alium metietur. 

Sint quotlibet numeri deinceps proportionales Ay 

By r, jjy Ey et A nume- 
^, ^^^^ ^ ^^ metiatur. dico, 

B\ . j^g alium quidem ullum 

^^ ' alium mensurum esse. 

ji , i^jj^ Y^QQ quidem mani- 

Ei 1 festum est, numeros j4, 5, 

Zi 1 r, ^f E deinceps inter se 

H\ : non metiri. nam j4 nume- 

^, rum B non metitur. dico, 

ne alium quidem uUum 
alium mensurum esse. nam si fieri potest, A nume- 
rum r metiatur. et quot sunt ^, 5, JT, tot sumantur 
minimi numeri eorum, qui eandem ac ^, ^, JT ratio- 
nem habent Z, H, S [VII, 33]. et quoniam Z, H, & 
in eadem ratione sunt ac ^, ^, JT, et multitudo nu- 
merorum ^, B, F aequalis est multitudini numerorum 
Z, H, 0, ex aequo erit A:r=Z:e [VII, 14]. et 
quoniam est ^ : 5 = Z : if, et ^ numerum B non me- 



1) Nam H : K^' H : 9 X G : K et H : 9 ^ T : E, 
e : K^ d :Z, 

Euolidoi, edd. Heiberg et Menge. U 19 



i 



290 ETOIXEISiS Ti'. 

//" ovx aga fiovttg icriv 6 Z' ^ ya(f (lovas jcavta 
agi&ftov (iEZpit. xai EiOiv ot Z, & npraroi JtQos dXXTJ- 
Aowg [ovdl 6 Z aga zov & (letQct:]. xal iariv as o 
Z Wpog tov @, ovzcos o A wpog xhv f- oiiS\ o A 
5 apa xov r (icrQEL 6[i.oio3g di} SEi^ofitv, ori, ovSl 
ttk\o£ ovSeXs ovdiva [lETQ^eEr ojieq iSEi SEC^at. 

r. 

'Eav axiiv 6no0oi.ovv agt&fiol [e|^s] o 
Xoj^ov, 6 dl nQiaTos xov Ee%tttov fitv^j}, xal 
10 Tov dfwTipoi' (lEtp^ffft. 

"Edztaottv O71O0OIOVV agi&nol i^^s avaXoyov olh 
A, B, r, /i, o 8% A Tov A (tETQaita' ^iyiOy 5« 
6 A Toi' B ftEtgst, 

Ei yttQ ov (lETQEt 6 A TQV B, ovSi aJiXos 
15 SeIs ovSiva (letQi^OEf (letQeC Se 6 A rov ^. [teri 
ttQU xal 6 A zbv B' otceq iSEt SEt^ai. 



Eav Svo api&[iav (leta^v xaTa to ffuwcjjjf 
avdXoyov i[t.nizTin0i.v aQi9(i,o£, oaot eis "Et)- 
j Torjg (letaiv xata t6 ovvsxes dvaloyov i(i- 
Tiiatoviftv aQtQfioi, zoOovzoi xal aig tovg x6v 
ttvtov loyov ijtovTttff [avtots] (lEzaiv xaxa %i. 
awtx^S ttvdXoyov i^itEBovvttti. 

Avo yag dQi9(imv zav A, B (tEza^v xctta 
i ovvExis dvdkoyov iiixiaxiTiaaav dQi&(iol oC F, 



S. fiETper afi&jiov Vqn, jiai f/oiv] om. q>. 3, 

o Z (cpcr TOV & (tttpei'] om., P. 6. (iExgei BV<p. 

li^f] om. P. 9, fox'""'] in '»*- ^- 1". diVTCQov] i2 

raa. V. 12. ^bC} om. ip. 14. ov] fi^ BVtp. 16. Pp«^ 



f ELEMENTORUM LIBER VUI. 291 

tituT, ne Z quidem aumerum H metitur [VII def. 20]. 
itaque Z unitas doq est; nam mitas omnem numerum 
metitur. et Z, & inter ae primi Bunt [prop. IH]. et 
eat Z ; @ = ^ : r. itaque [VXI def. 20] ne A quidem 
numerum F metitur, similiter demonstrabimua, ne 
alium quidem uUum alium mensurum esse; quod erat 
demonstrandum. 

vn. 

8i quotlibet numeri deinceps proportionalea sunt, 
et primuB ultimum metitur, etiam aecundum metitur. 
A\ — I Sint qaotlibet numeri deinceps 

Bi 1 proportionales j4, B, F, ^J, et A nu- 

Ti — f merum z/ metiatur. dico, A etiam 

j\ 1 numerum B metiri. 

nam si A numerum B nou metitur, ne alius qui- 
dem ullus alium metietur [prop. VI]. metitur autem 
A numerum ^. ergo A etiam numerum B metitur; 
quod erat demonstrandum. 

VIII. 
Si inter duos numeros secundum proportloiiem 
continuam numeri aliquot interponuatur, quot inter 
eos secundum proportionem continuam interponuntur 
numeri, totidem etiam inter eoa, qui eandem ratio- 
nem babent, aecundum proportionem continuam inter- 



Nam inter duoa numeros A, B secundum propor- 
tionem continuam numeri aliquot F, J interponautur 

fiGip^'«ci add. V^: wiitij atonov' vnoKeiiut yag o A lov J 
/texfeiv; idem B mg. m. 2, 82. avtoie] om. P. 26. r] 



1, TO- l| 

4- ot 



292 STOIXEltiN 15'. 

xal xmairia^m mg h A repog zov B, ovtojs o E 
:iiQog xov Z" Aeyro, oti otfo( £('s zovg ^, B (ifTtt^v 
xara ro ffwe^is avakoyov ifiTismmxaeiv aQt&fii 
aovToi xal tlq rovg E, Z (isTaJ^ii xmet xo evvi 

b ivdXoyov Ifimeovvtai. 

"Oaoi yuQ elot- r*5 nXri&st of A, B, F, ^, roai 
TOi siXtjqi&maav iXcfj^iaroL aQi&(tol rmv tov 
ki;yov iiovTtov xolg A, F, A, B ot li, &, K, A' of 
aga axgoi ttvzav oC H, A niprarat Jtpos «AAijAi 
10 slalv. Jtal i%sX ot A, r, A, B toCg H, ©, K 
za «vrp Xoya sloCv, xaC iati.v iaov z6 wi^^&os 
A, r, A, B rra ^iXij&si rmv H, &, K, A, di taov 
ietlv mg 6 A Jtpog rov B, ovzmg 6 H ffpog 
mg di 6 A Jtpog rov B, ovtiag 6 E XQog tov Z" xal 
15 mg apa o H Jipog rov A, ovzag 6 E n;p6g rov Z. 
oC di H, A affmtoi, ol S^ %Qmzoi xaX eXdj^tatoi, ol 
ds ildxtezot aQi&itol iiizffovat tovg tiiv avtov Xoyov 
sxovtag iedxis o ts {isC^mv rov fisC^ova xal 6 i^da- 
emv zov ildeeova, zovriaztv o rs ijyovftsvog rov 

lyovfisvov xttX 6 saofLsvog zov iaofisvov. laaxig 
aqa 6 H tov E ftErpst xaX 6 A zbv Z. oadxts S^ 
6 H tov E ftstQsl, zoeavtdxtg xaX ixatspos zmv 0, 
K ExdtsQov zmv M, N fiszQsCtm- ot H, ®, K, A aQO 
zovg E, M, N, Z iedxig fiErQovatv. ot H, &, K, A 

6 dqa Totg E, M, Y, Z iv za avra Xoym siaiv. aXXa 
ot H, ®, K, A zotg A, T, A, B iv r© avtm Ad; 



3. tol Tot ip. 6. [Caiy B. 7, at hdxiazi 
r, d, S] B, r, J BV^. oE] corr. es tovg m. 

of] om. P. 10. tlalv'] tlo,' V<p. ■ul i-itei — 11: tlaiv] 

om. w. 10. r] in ras. B, pOBt ras. 1 litfc. V, 11, elat V, 
13. zov Al A B. 18. fjovrns avToh BVqi. 1«. ze'\ om. P. 



1 



Ki 



ELEMENTORUM LIBER VIll. 293 

Ei ' et fiat ^ : B = E : Z. 

-Mi- 1 dico, quot inter J, R 

—' secundum proportionem 

— > continuam interponan- 

tur mimeri, totidem 

etiam inter E, Z secun- 

K '-' dum proportionem con- 

A\ 1 tinuam interpositum iri. 

nam quot aunt uumero A, B, F, ^, totidem su- 
mantur numeri minimi eorum, qui eandem rationem 
habent ac A, T, ^, B [VH, 33] H, 0, K, A. itaque 
extrerai corum H, A inter se primi sunt fprop. ITI]. 
et quoniam A, F, jJ, B et H, @, K, A in eadem ra- 
tione Buut, et multitudo numecorum A, F, j^, B mul- 
titudini numerorum H, ®, K, A aequalis est, ex aequo 
erit [Vn, 14] A:B = H: A. uerum ^ : B = £: Z. 
quare etiam H : A = E: Z. sed H, A primi sunt, 
primi autem etiam minimi [VII, 21], minimi autem 
numeri eos, qui eandem rationem habent, aequaliter 
metiuntur, maior maiorem et minor minorem [VII, 20], 
h. e. praecedens praecedentem et sequens sequentem. 
itaque H numerum E et A numerum Z aequaliter 
metitur. iam quoties H nnmerum E metitur, toties 
wterque 0, K utrumque M, N metiatur, itaque H, &, 
, A numeros E, M, N, Z aequaliter metiuntur. ita- 
^ue H, &, K, A et B, M, N, Z in eadem ratione 
mt [Vn def. 20]. uerum H, &, K, A et A, F, ^, B 




294 ETOIXEIflN ij', 

sieiv *al ot J, r, ^, B aQa rofg E, M, N, Z iv zS' 
amm koya ileiv. ot b^ A, T, /i, B i^^g avakoyov 
dniV xal oi E, M, N, Z &qu il^g avdioyov iiaiv. 
oflot cpa tCs Tovg A, B neta^v xara ro Uwex^S ^'"u- 
5 Ao^-oi' iiinmrmxaaiv aQi&fioi. roGovTot xal ais ''^ws 
E, Z (itra^v xara to avvsxig avakoyov iftJiexrmHaaiv 



'Eav dvo aQi&fioi stQmTOt itQog dXXi^Xovs 

10 aatv, xal tls avrovg iitTa^v «ata z6 avvsx'^? 
avdhoYov ifi.aiitTaaiv aQi9fioi, oeot sig av- 
loug (tfca%v xura t6 avvEX^S avaXo-yov i(i- 
itinzovaiv aQi&fioi, roooiJroi xal sxazeQov av- 
rmv xal [lovaSog [leTaiv xara t6 avvExeg avd- 

iG Aoj^ov Efi%BaovvTai,. 

"Earmaav Svo aQi&fiol agtorot «pog dlXiji.ovs o{ 
A, B, xal tlg avxovg ^tra^v xaza t6 awExts dvd- 
koyov ifi.at7tTdT<aaav oi V, z/, xal ixxeia&m t/ E p,o- 
vdg' Xayo, OTi oaot lig tous A, B titrtt^v xara t6 

20 avvEyis avdXayQv i^itEnxmxaaiv aQid)io£, TOtfovroi 
xal ixaxi^ov tcov A, B xal r^g (lovddog iitra^v xark 
t6 avvE%ES dvdXoyov ifijttaovvzai. 

ElXrj(p%'faersv yaQ Svo ftiw uQi^^^ol iXdxtGzot ^ 
tk! rmv A, F, ^, B Xoyp ovrtg oC Z, H, rptts di ot 

26 0, K, A, xal del E^ijg ivl jtXEiovg, Emg av tOov yivjb 
rat t6 JtX^Qog avrmv Tra aX-^&tt Tmv A, F, /t, J 
tiXrm>^mGav , xai laraaav ot M, N, ^, 0. ^avEM 



om. P. xai ot — 2: Idyra clalv} mg. i 
1^01»] (prius) elai Yip. 'lO. roo» PV91 



J ELEMENTORUM LIBER VIU. 295 

ia eadem ratione aunt. quare etiam ^, F, ^, B et 
E, M, N, Z in eadem ratioiie sniit. uerum ^, F, jJ, B 
deinceps proportionalea Bunt, quare etiam E, M, N. Z 
deincepa proportionales sunt. ergo quot inter A, B 
secundum proportionem coiitinuam interpositi sunt 
numeri, totidem etiam iuter E, Z secundum propor- 
tionem continuam iuterpositi sunt numeri; quod erat 
demonstrandum. 

IX. 

Si duo numeri inter se primi sust et inter eos 
aecnndum proportionem continnam interponuntur nu- 
meri aliquot, quot inter eoa aecundum proportionem 
continuam interponuntur numeri, totidem etiam inter 
singulos et unitatem Beeundum proportionem conti- 
nuam interponentur. 

Sint duo numeri inter se primi A, B, et inter eos 
secundum proportionem continuam interponantur F, 
^, et ponatur unitas E. dico, quot inter A, B secun- 
dum proportionem continuam interponantur numeri, 
totidem etiam inter singulos A, B et unitatem aecun- 
dum proportionem coutinuam interpositum iri. 

sumantur enim duo numeri minimi in ratione A, 
r, ^, B numerorum Z, H, tres autem &, K, A et sem- 
per deincepa uno plures, douec fiat multitudo eorum 
multitudini numerorum A, V, ^, B aequalis [prop. II], 
Samantur et sint M, N, S, O. manifestum igitur 






ppiO^^oi oaoi] in rae. m. I B. 12. f^tnAncaoiv P. 
#B«o£«] ag^K iinaiv Theon (BVip). 24. rav] coir. ei r 



296 STOIXEIiiN 71. 

dij, oTt o ft^v Z iaxrtbv n.oi.kajtXaetdeag rdi 
soiTjxEv, tov di @ noklanlaaiaoas tbv M Jteaoirjxfv, 
Httl 6 H iavTov fiev nolXaTcXaaid^ag zbv A iteTtoifjxev, 
Tov dh A 7ioXl.a%XaSiuGai xbv O acaoitjxev. xal ixii 
6 oC M, N, S, O ildxi.eToC Biot. rmv tov avrbv koyov 
ixovratv rorg Z, H, eiffl 61 xal oC A, F, ^, B iXd- 
Xiotot tmv tbv avtbv Jioyoi' iiovttav TOlq Z, H, xai 
idTiv [aov t6 jtkrl&og irow M, N, S, O va xi^&ii 
tav A, r, z/, B, fxastog apo t<av M, N, S, O ixaatto 

10 tav A, r, J, B faos ietiv ieog &Qa iatlv 6 (thv M 
rro A, b Sb O tra B. xal insi 6 Z iavtbv jzoXXa- 
nXaetdaaq tbv & nmoiijXEV, b Z aga toi' @ (istqU 
xara tas iv xa Z liOvdSag. fiet^tff di xal fj E ftovds 
roi' Z xtttd Tag iv ccvtta [lovadag' iaaxig a^ix i^ E 

15 (lovaq zbv Z KQL&fibv (istQsi xal b Z zbv ®. SaTiv 
ttQK mg 7) E [lovdg jrpog zbv Z dQi&(i6v, ovzas o Z 
Tcpbs rbv &. adXi.v, iasl 6 Z rov & aoXXa7tXa0td- 
eag tbv M Mitoitixtv, o ® aga tbv M (lEZQst xara 
Tag iv t^ Z tiovddas. fifTpei Se xal ij E (lovag 

20 tbv Z dQi,9(i6v xazd tks iv avTp fiovtidas' iddxig 
«pa i) E jiovdg Tov Z dgtd-fiav iietqbC xal 6 tbv 
M. iativ wpa tog ?j E (lovdg ngbg zbv Z txgt&fiov, 
ovzmg 6 & XQog toi' M. iSsix&vi S^k xal rag ij £ 
^ovaq a^bs tbv Z dqt&yiov, ovTtog 6 Z Ttqbg rov 



1. jteao^ijKE Ttp. 2. newD^ijxf Tqi. 3. jtfreo/^Me Vql. 
4. Ti[-7io(ri*E Yw. 6. tlaiv P. 6. Z, H] H, Z BVtp. ' 
tf^j- B. 7. 10*] corr. ei r£» m. 1 P. Z, H] H, Z B 
E, Z P. 10. ftros] (prina) corr. ex taov m. rec, P. 

ZJ eras. V. 13. tco Z] mSiro V(p, ica Z aupra 
opo] fii <p. 21. «] e corr.'V; E P. 22. ii«] supra 
84. xgog] (priaa) Bnpra m. S B. 



1 



ipra m. 1 ffJH 



ELEMENTORUM LIBER Vni. 297 

est, esse Z>C.Z^%, ZxG^ M, HxH= A, 

HxA = [prop. 

n coroU.]. et quoni- 

' am M, N, IS, O mini- 

d\ 1 ^i 1 mi sunt eorum, qui 

jBi 1 eandem rationem ha- 

^, j^^ , bent ac Z, H, uerum 

etiam ^, JT, J, B 



A\ 1 01 

T\ 1 K 



H 



minimi sunt ^orum, 
I ^^. ggj^j^jj^ ratio- 

0\ 1 nem habent ac Z, H 

[prop. 111] , et mul- 
titudo numerorum M, N, S, O multitudini nume- 
rorum ^i, F, d, B aequalis est, singuli M, N, S, O 
singulis j4, r, jd, B aequales sunt. itaque M = A, 
O = B. et quoniam ZxZ = S, numerus Z nume- 
rum ® secundum unitates numeri Z metitur [VII def. 
15]. uerum etiam unitas E numerum Z secundum 
unitates ipsius metitur. itaque unitas E numerum Z 
et Z numerum ® aequaliter metitur. itaque 

E:Z = Z:® [VII def. 20]. 
rursus quoniam Zx® = M, numerus ® numerum 
M secundum unitates numeri Z metitur [Vll def. 15]. 
uerum etiam unitas E numerum Z secundum unitates 
ipsius metitur. itaque E unitas numerum Z et & 
numerum M aequaliter metitur. quare 

^:Z = 0:M[VII def. 20]. 
demonstrauimus autem, esse etiam E : Z = Z : ®, 



298 rroiXEiQN .j'. 

xal mg rqu r) E [lovag ngos tov Z agi&fiov, ovtmg 
b Z TTpo? xov @ Xttl 6 & jrpog xov M, /"tfog di o 
M ta A' lariv apa d>s f] E (iovag Kpog t.ov Z 
Bpt&^iiov, owiras o Z irpog xov & xal o & TtQtg rov 
6 ,^. tftK TK ttvtK Si] xal as ij E fiovas n^pdg tov 
H dQt&fiov, ovzas 6 H npog rov ^ x«l o ^ «p6; 
TOi' B. offof KpK a'ff Towg y/, B fi£T«£ii xora t6 (Tiii'- 
iXSS avdXoyov ifinsnziaKaeiv «pi&fioi. Toffot/rot xiu 
ixai^ffov tmv A, B xal (lovddos T^g E ^£Tai,v xaztt 
10 t6 Owixis dvdXoyov iftJCtntioxaOiv KQt&(J.ot' oni(/ 
16 Bt Sel^at. 



'Eav Svo ttpi&fimv ixarigov xal iiovaSog 
jtEta^v xata th evvBxis avdXoyov ifiaixtcottiv 

16 di}i9[ioi, aSot ixatsQov avtmv xal {lovaSos fte- 
ta^v xaru t6 Ovvexhs dvdkoyov i^nCnrovatv 
dQi9fioi, ToOoutot xal elg aijTOiig fiETKgii xata 
t6 avvsxks ttvdXoyov ifL^BBovVTaL. 

^vo yap dQid^fimv tiav A, B xal fiovdSos t^s f 

20 [iBTa^v xata rb <fuv£j;ig dvd2.oyoi' limiTcriTioaav apt^ 
fiol oZ ts ^, E xal ot Z, H' liya, on oOot sxati- 
0OV Tcov j4, B xal [lovdSoe t^g F ftsra^v xata ro 
avvex^S dvikoyov i{i7tS7Ctiaxtt0tv aQi&ftoi, couovTtH. 
xal elg TOvg A, B ^tTa%v xata ro evvsxkg dvdkoyi 

25 ifinsaovvtai. 



2. icgbg tov M — i: TtQOi tov J] adil. m. 2 B; -eed xgif 
TOV A Uq. 4 eticnD in textu Bunt a. m. 1. % looe Si o M x^ 
A] b 6i M (liij ip) im .4 laiiv rffog BVqo; in V haec nerbil 
et seq. ad ir[iD£ lov A liu. 4 in mg. sant m. 2. 3. i]] 
ei D ^. 13. «xaMpoti] om. Theon (BV9). )6. eJ^s 
Ta|ii Theon (BVq!). 16. to] om. V. 18. aWloyov] 

2 B, om. V9. 



ro 




ELEMENTORUM LIBER Vm. 299 

quare etiam E: Z = Z:S ^ & : M. uerum M = ^. 
itaque erit E : Z = Z : @ = & : A. eadem de causa 
etiam E : H = H: A = A : B. ergo quot inter A^ B 
seeundum proportionem continuam interpositi sunt 
numeri, totidem etiam inter singulos A, B et uni- 
tatem E secundum proportionem continuam inter- 
positi sunt numeri; quod erat demonstrandum. 

X. 

Si inter duos numeros^) et unitatem secundum 
proportionem continuam numeri aliquot interpositi 
sunt^ quot inter singulos et unitatem secundum pro- 
portionem continuam interpositi sunt numeri, totidem 
etiam inter ipsos secundum proportionem continuam 
interponentur. 

^ . Nam inter duos numeros 

_, ^, 5 et unitatem F secundum 

proportionem continuam iuter- 

p, ponantur numeri ^, JB et Z, 

, „ I 10 H. dico, quot inter singulos 

„. A, B et unitatem F secundum 

j j^ proportionem continuam iuter- 

. positi sint numeri, totidem eti- 

am inter A, B secundum pro- 
portionem continuam interpositum iri. 



1) Scripturam codicis P lin. 13 {srtccxiQOv) etiam Cam- 
panns habuisse uidetur; apud eum enim VIII, ^IO ita lecimus: 
si inter utrumque eorum et unitatem quotlibet numen con- 
tinua proportionalitate ceoiderint, ambobus numeris totidem 
continua proportionalitate interesse necesse est. 



300 ETOISE1.QN ij: 

'O ^ yap xQv Z 7ioli.anlaatdeus tov & itouita, 
fXKTfpos li^ Ttov ^, Z thv © nollajtlacideas txdte- 
Qov rmv K, A 'noitCra. 

KaX inti iettv ms ii F (tovag ^Qog tbv ^ aptO'- 
h /tciv, ovxfog 6 A JTpos toi' E, iadxis «9« V ^ (lovai 
Tov A api&fiov (lerptt xal 6 ^J rov E. ij dh P (lo- 
vKg Tov ^ dgt&fiov iisrpsi xata tdg iv ra /i (io- 
vdSas' xal 6 A apa api'9^os tov E (ittpBt xara TKg 
(V TM z/ (lovddas' o ^ apa iavtov nolXaiiXaaidaixs 

10 Tov E TtmoCrixBv. jidktv. insi iartv tog 7) r [fiovag] 
XQog tov ^ dgt&ftov, ouTOg 6 E n:pos rbv j1, iadxig 
aptt 1} r (lovdg tov ^i dpiQ^iiov fistQti xal 6 E tov 
A. ii S% r fiovdg tbv A dffi&[ibv iitt^et xatct tdg 
iv ip ^ fiovdSas' xal 6 E ap« toi' A fisteti xata 

if) tdg iv ra /i fiovdSag' 6 ^ dpa tov E nokkankaetA 
eag T^bv A ■xenoir^xsv. Sid rd avrd Si} xal 
2 ittvtbv noXlajiXaaidaas rbv H ■n.snoCtixsv , tov ! 
H noXXankaatdaas tbv B nsnotrixsv. xal inel 6 d 
savtbv ftiv nolXanXaGtdaag rbv E nenoiijxsv, 

20 Si Z noXXanXaatdaas rbv & nsnoCrjXSV, Seriv aga 
fog 6 •^ irpog Tov Z, ovtag 6 E xpbs i 
Ttt Rvtd Sij xcei rag 6 ^ npbg rov Z, ovtms & j 
ngbg rbv H. xal mg dpa 6 E npbg rbv &, t 
o @ 3ipog Toj' H. ndXiv, insl b z/ ixdtsQov tmv 

26 E, & nokXanXaaiaaag ixaregov tmv A, K neaoiijxsv, 
lativ aga tbg o E nQog rhv &, ovrtog i 
Tov K. aKk' as 6 E «pos tov &, outeis 6 ^ Jti^ 
rbv Z' xal ag £p(t o jd nQog tbv Z, ovTog 6 ^ 

4. iaciv] supia m. 1 V. 8. «al 6 d apo — 9 
mg. m. 1 Pip. 8. Sga] oto. B. dgi^fioi] om, 
ni-noCti^f Vq>. iiavds] om. P. 12. C] e corr. 



xaftt 

'", ^ 

i J 

TOV 



Kk. 



ELEMENTORUM LIBER Vni. 301 

sit enim -d^xZ = ®, ^X® = Jir, Zx0 = ^. 
et quoniam est F: J — jd i Ey unitas JT numerum z/ 
et jd numerum E aequaliter metitur [VII def. 20]. 
uerum unitas F numerum z/ secundum unitates nu- 
meri z/ metitur. quare etiam numerus z/ numerum 
E metitur secundum unitates numeri z/. itaque 
zf X ^ = E. rursus quoniam est F : ^ = E : j^, 
unitas F numerum ^ et ^ numerum A aequaliter 
metitur. uerum unitas F numerum z/ secundum uni- 
tates numeri z/ metitur. quare etiam E numerum 
ji secundum unitates numeri J metitur. itaque 
J X E= A. eadem de causa etiam Zx Z = H 
et Zx H = B. et quoniam jd X d = E et 

J XZ = ®, erit [VH, 11] J :Z = E:®. 
eadem de causa erit etiam J :Z = S :H [VII, 18].^) 
quare etiam E:0 = ®: H. rursus quoniam 
JXE = Aet^xe = K, erit E:& = A : K 
[VII, 17]. uerum E: ® = jJ : Z. quare etiam 

J:Z = A:K. 



1) Cum habeamus d x Z ts» B et Z x Z =^ H, proprie 
citanda est YII, 18, non YII, 17, ut iu praecedenti ratio- 
cinatione; sed cum J x Z ^^ Z x J (VII, 16), adparet, Eu- 
clidem sine errore dicere posse lin. 21 sq.: dia ra avra. 



iaccTiig — 12: rov A] bis V (corr.), g>. 14. xal 6 £ — 16: 

uovddag] mg. m. 1 P. 14. ^] in ras. m. 1 B. 16. nsnoiriKS 
Vqp. 17. nsnoirfas Yqp. 18. noXlaaulicag qp. 19. ns- 

9roiV}X6 Yqp. 24. twv E — 25: s-KatBQOv] mg. m. 1 P. 

26. xov A, H (f. 27. dXXd P. 



I 330 ETOIXEIiiN .)'. 

'O /! yft^» xhv Z TtoXXa.xXaeMOa^ tov 6 ; 

pOK tmv A', ,4 ■noitita. 

Kal iitei ie-riv an; »j T ftovag jcpij tow ^ 
h [lov^ ovzas /i nrpog tdi' £, iadxLg affu ^ i 
rov ^ aptftfiov iiitget xal 6 ^ t^v K ^ di 
VRS Tov z/ aQi9n6v fLSTifEt xaia t^$ ^v Tfo 
va6ag' xal 6 -^ «pa api9nog rhv E fiBxget x 
iv T^ ^ (lovaSas' 6 ^ aga iavrov noiUinril 

10 tov E itsaoitjxiv. TiaXiv. inei iexLv as ^ P 
itQog %ov ^ aQi&^ov, ovTmg o E XQOg i^ov -^. 
Spa ^ r iiovccg tov ^ uQi^^v ftcrpcr xat h 
A. ij Si r iiovag tiW ^ aQi&fiov fttzQtC xi 
iv ra ^ jiovadag' xal 6 B aga tov ji fiET^ 

15 tag iv rra ^ (lovudag- o /i aga rbv E xoXXa. 
tlas Tov A ne^oiYjXiv. 6ia ta «vra Sij xaX 
Z ittirtov noXlaaXaCiaaag tov H TtsiioCrfxev , 
H xoXXaxXaetaSag tov B «c^toitixsv. xal in 
iavTov ftiv xoXkazXaeia^ag tov E iteaoitjxi 

20 Sh Z nolXanXaeideas ti>i' © itsaoi^xev, iOt 
ag 6 ^ Jtpog Toi/ Z, oifrrog o E Jtgog tov i 
TK Kirttt d^ xal ag o J ngog thv Z, ottT* 
srpog Toi' H. xal wg Kpa 6 £ Kpos tov & 
6 & jrpos tov H. nakiv, inei 6 A ixdreQ' 

26 E, & xoXXaTlXaeidaag txdtBQOV ttav A, K aej 
isriv apa a>s o E nQog xov &, ovxmg 6 . 
tov K. aXX' ag o E npiis tov @, ovtwg o . 
Tov Z' xal mg apa o z/ ngog thv Z, ovTa 

i. laxiv] aupra ni. 1 V. 8, *a\ 6 d iga — 9: ( 
mg. ra. 1 Pqj, 8, uga] om. B, Hpi&fios] om. Vi 
jitJio/Tjxs Vy. fiovas] oin. P. 12. P] e corr, V 



302 



rroixEaaN i; 




^ipoj Tov K. naiiv, iitil ixiittifai tfflw ^. 
jtoi^aalttatdatts ixatsQov tcav A', A xBTcoiijxcv, iettv 
ttQu as l) 'd JiQOS tbv Z, otlrras 6 K xpog toi/ v^. 
kAA cos q zl stQoq tov Z, ouiog o ..4 xgog lor £' 
6 xal mg «pa o A ngog tov K, o^rog 6 K itpog rov 
^. Iti iatl 6 Z £xarf£)ov tav &, H nalXaalaOtecSCLi 
ixdtfQov ttav A, B mxoirjxfv, sati.v «(/a tas 
TCQog tov H, ovteog 6 A npbg tbv B. mg iS^ 
^i^og Tov H, ovtcog b /S agog rbv Z' xal mg a| 

10 o i^ npog tbv Z, ovztag 6 A ngbg tbv 3. iSeijpii 
dh xal mg b jd aQog tbv Z, ovtag o te A xpbg xbv 
K «al 6 K npbg tov A' xat mg uQa 6 A 3tQbs tbv 
Kf ovtaig b K XQog tov A xal b A npo? tov B. ot 
A, K, A, B aga xata ro fluvfxtff *§^S tiaiv avdXoyttv. 

16 offot dpa ixazdpav tiov A, B xal t^g F [lovaSog )1.e- 
taiii xata tb awexis «vkAoj-ov iiinCntovaiv dpi&fioi, 
lOBouTot xal sig lows A, B fitta^v xata tb ewey^ 
iimeaovvxai' oneQ iSsi 8Bl%ai. 



iag 

1 



kovov iativ I 



SinXaaio 



■pi&^fiav clg fiiaos dv£ 
Tstpdyoivog «Qa 
Xoyov l%Ei ijWBl 



7} nXtvga TtQoq tijv %Ie 

"Eataaav TstQdymvoi aQt&fiol oC A, B, xal za 

26 ft^v A nXsvQa sata 6 F, tov Sc B 6 A' Xsya, Sl 

zmv A, B stg (liaog dvdXoyov iartv dQt&[i6g, xal J 

A npbg tbv B StitXa6iova Xoyav i^ei ijjrEp o F ^Qm 

10 V jd. 



Z] Z, J B. 3. Z] i 
IS. xal as Stia - 



1 ^dhv, deleto xai P. ^ 

10. ISeix^^ ^^] ™g' 9- 



ELEMEKTORUM LIBER Vni. 



303 



rursuB quoniam ^ X & = K ei Z X & = ji, erit 
jd : Z ^ K : A [VII, 18]. uerum ^ i Z = A : K. 
quare etiam A : K — K : A. praeterea quooiam 
Zx@=^AetZxH=B, erit [YU,n^&: H= A:B. 
uerum @ : H = iJ i Z. quare etiam ^ : Z = A : B. 
demonstrauimus autem, esse etiam 

^:Z= A:K = K:A. 
itaqne eiit A : K = K i A ^ A : B. itaque A, K, A,B 
deinceps in contiuua proportione sunt. quot igitur 
inter ainguloa A, B ei T unitatem secundum pro- 
portionem continuam interpouuntur numeri, totidem 
etiam inter A, B deinceps interponentur; quod erat 
demonstrandum. 

XI. 

Inter duos numeros quadratos unus medius est 
proportionalis numerus, et quadratus ad quadratum 
diiplicatam rationem habet qnam latus ad latus. 

Sint nnmeri quadrati A, B, et numeri A latua 
sit r, Bumeri autem B latus ^. dico, inter A, B 




304 ETOIXEmN 11 '. 

'O r yaQ rhv ^ aoXlanlaeiaeas tov . 
xal ijitl Tiz^ayavos iexiv 6 A, nXivpa 6i avrov 
ietiv 6 r, 6 r apa iavtov noXkankaoiaaag tov A 
nsnoifixtv. Sia ra avza Sr] xal 6 ^ EatfTo»' ^xoAAa- 
5 jilaeidaas rov B ntnoCijxfv. insl ovv 6 r ixdrEfiov 
rwv r, ^ noXlanlaa^daag ixdtBQOv rtov A, E si- 
xoitjxev, e6ztv «po: rag 6 F ffpog tov z/, owrog o 
ji xpog loJ' E. did tis. avtd Sij xal tos 6 Jf «pog 
zov iJ, ovras 6 B repog tov B. xal mg «por i 

10 Kpog Tov E, otTTrog 6 £ ropog tov S. rtov A, B i 
ets fi^tfog «vadoyov ^ffTiv api^jiog. 

jiiym d^, ort xal 6 .4 npog rov B dutXaeioi^ 
ioyov ij^ei tJjisq 6 F itgos rov jJ. insl yorp 
dffi9(iol dvdXoyov tCeiv oC A, E, B, o A apa , 

16 Tov jB SiTcXaeiova Xiyov ixn ijnfp o A %Qas i 
E. dis S\ u A jrpos Tov E, ovtcag 6 J" jrpoe tov 
A. A aqa Jipos roi' B dmXaeiovK Aoyoi' i^^jei ^jTEp 
^ r nXcvQtt ngog ti]V .d' OJTep edEi Set^ai, m 



wpos 
e o A 

B&nM 

ig tA^* 



20 ,^t!o xu^rar dpi&^^v Svo ftEOoi ai/a^ojjov 

eiaiv dgi&[ioi, xal 6 xv^og repog tov xvfiov 
rgmXaaiova Xoyov e%ei- ijntQ ^ nXsvpd ngog 
fijv nXivgdv. 

"EeTnaaav xv^oi. aQi&itol oC A, B xal rov (tiv 

ii5 xXBVQd iatta r, tov Si B 6 A' Xiya, o« rav 
B Svo iiieoi dvdXoyov eleiv dQiQ^ioi, xaX b A X{ 
Tov B TQiTcXaeiova Xoyov i^si jJwEp o f «ifog vov 



1. yoQ^ m. 



2 B, post ras._ 1 litt. V.^ 
iri v.ai] P; iidliv irctl o 
iao£i}r.rv, o di J iavtbv 



4. jrejto/ijKe Vd 
r TOv A nollemta 
wlXaaXa.ei.daaz i * 



m ELEMENTORUM LIBEH VIU. 305 

sit enim Fx zl = E. et quoniain quadratus est A 
et latus eina T, erit F x T ■= ^. eadem de causa 
etiam ^ X ^ = B. iam quoniam rxr= A et 
r X ^ = £, erit r:A^A:E [VII, 17]. eadem 
de causa^) erit etiam F : A = E : B. quare etiam 
A : E = E : B. ergo inter A, B imus medius eat 
proportionalis numerus. 

lam dico, esse etiam A : B = T* : ^*. uam quo- 
niam trea numeri proportionales sunt A, E, B, erit 
A : B = A^ : E^ [V def. 9]. uerum A : E = T : A. 
itaqne A : B = I^ : A^; quod erat demonstrandum, 

XII. 
Inter duos cubos uumeros duo medii proportio- 
Dsles sunt numeri, et cubus ad cubum triplicatam 
rationem habet quam latus ad latus. 

_^i _^ ,^1 Sint cubi numeri 

., A, B, et latus nu- 
meri A sit F, numeri 
autem A. dico, 
iuter A, B duos me- 
dios proportionales esse numeros, st esse AiB^r^: ^*. 

1) Nam r X iJ = E et.iJXiJ = B. itarine proportio 
illa proprie per VII, 18 (non VII, 17) efficitui. sed cfr. 
p. .100, 21 aq. et p. 301 not. uerba lin. 8 interpolata etdEua 
ipsa oratioois forma (tva xal lov aviov) redargunntnr. 



B Kfno^ijKCV {ictJioitjTie Vip), Svo 6^ dfi&fiol af r*, d ¥vit Kai 
Toc avthv lav J TiaHaTcXaaiaaai^Tee Toug E, B ntitotnnuoiir ' 
iativ S^a Theon{BVqj). 9. Post B ftdd. Theon: dXl io$ 6 
r irpos 10* J, oftojs A Ttffhs tov E {BVrp). 10. t(ov] 

Too in raB. oomp. V, 11. api&fios o E Thoon (BVy). 18. 
^ xlev^av Vcp. 20. niaove P, corr. m. reo. 



■'l 1 


01- 


X 


z 




H 







L 



306 SITOIXEIiiN ^■. I 

'O yttQ r tavTov [ilv %okkax).c((iideas tbu E xoitino, 
101- de A jiollaai.aeiu0as tov Z xoiBhm, o dh ^ iavtov 
nokXaalKaiiiaas roc H notciia, ixcrtpog dh ztov F, 4 
xiiv Z itoXXaTiXaa\,aaas exfitipov rmv &, K itouirm. 
Kal intl xw^os iatlv 6 ^, vtltvpa 6i avtov 6 
r, xal r iambv jtollaxXttai,daa$ tbv E xsitoCtjxev, 
o r aga iavtbv filv jtolKajtlaatdaas rbv E ttgsoiV 
xBv, tbv dh E TtolKanXaaidaas tbv A iiEaoCr}Xtv. 
8itt ta avta Sij xal 6 ^ ittvrbv (liv aoi.Xa3tiaS1.doas 

10 tbv H 7t£3toit]xiv , tbv Si H jtoXXaTtXaettiaas tov S 
Ttt^toirjXiv. xal iatl 6 r ixdtsgov tav V, ^J JtoXAit- 
TtXaaidaas ixdtSQOi' rwv E, Z aeitoirjxtv, iariv apa 
mg b r TtQos tov ^, oiJTOJg b E wpos rbv Z. dta 
tu avza bi] xal ros o T srpos tbv ^, ovttas b Z 

16 «pos tbv H. ndhv, intl b F ixtttegov Ttov E, Z 
nolXanXaaidaas ixdtegov tmv A, ® atJtoii}xev, ionv 
Kp« as 6 E ngbs zbv Z, oifrojg 6 A ngbs rbv &. 
ms Si E 3Epog rbv Z, ovras F irpos rbv /i' 
xttl tog aga b F ngbs rbv ^, ovrms A npbg tbv 
0. ndXiv, iztl ixdtsQos rrai/ F, A xbv Z itoXkanXa- 
atdaas ixdrtgov rmv ®, K xctTtoi^^xev, iariv aga mi 
r %gbs tbv A, ovras 6 © rcpos rbv K. -jtdhv, 
ijtel b d ixdregov rav Z, H TCoXlttTtAttaiaaas ixdtt- 
Qov Twi' K, B :t£noir]xev , iattv aga d>s b Z irpof 
tbv H, ourras 6 K Jtpog toi' B. rog 6h i Z xgbs 
rbv H, ovrms o F agbs tbv z/' xal rag aga T X{ 
tbv Ji, ovtag o te A JtQbg tbv @ xal @ ff^og 
K ;(«! 6 K rcpog tbv B. rmv A, B «pa 8vq ptit 
avttXoyov tiaiv ot &, K. 

i. Z] eraa, V. 6. nenoiTjtit V93. 7. Jitxoii 
nEJiotijKE V ip. 10. jrtjio^ijxc Yip. 11. irmoAjn 



1 



ELEMENTORUM LIBER Vni. 307 

sit enim r X F = ^, Tx z^ = Z, z^ X z^ = i/, 
rxZ = @, ^XZ = K, et quoniam A cubus est, 
latus autem eius T et F X T = ^, erit rx r= E 
etrxE^^A, eadem de causa erit etiam z^ X z^ = if 
et^XiI=-B. etquoniamrxr=Eetrx^ = Z, 
erit r : ^ = E : Z [VII, 17]. eadem de causa erit 
etiam JT : z^ = Z : il [VII, 18].^) rursus quoniam 
rx^ = ^etrxZ = 0, eritE:Z = ^:© 
[VII, 17]. uerum E : Z = F : ^, quare etiam 
r : ^ = A : 0. rursus quoniam f X Z = @ et 
^ X Z = JT, erit [ VII, 18] r:^ = ®:K. rursus 
quoniam ^ X Z = K et ^ X H = B, erit 

Z:H=K:B [VH, 17]. 
uerum Z : H = F : ^, quare etiam 

r:^ = A:® = ®:K = K: B?) 
ergo inter A, B duo medii proportionales sunt ®, K, 



1) Nam r X z^ = Z et z^ X z^ = Jf ; u. p. 305 not. 

2) Euclides bic paullo breuior est, quam solet. sed re- 
cepto supplemento codicum deteriorum Im. 27 falsa illa effi- 
citur forma orationis, quam p. 302, 12 — 13 cum P sustulimus. 
cui ut mederetur, Augustus lin. 28 post prius K interposuit: co? 
aqa 6 A ngog tov G ovzcag o xb A n^oq xov K (I); ego malui 
codd. PB sequi. 



ovtmg — 18: nqog xov Z] m. 2 B. 20. I«6^] om. P. 26. 
B] Hqp. 27. Post d add. Vqp: ovxag 6 K nqog xov B' 

idsix^ri 8h %al mg o F nqog xov d\ idem B mg. m. 2. 
o Tf] Tfi 6 B. 28. Tcov] corr. ex xov V. 29. of] a^i<&'- 
fU)l ol B* 



20* 



ETOIXEIiiN Tj'. 



"■ 



yliym 6^, oti xal b A Trpos xov B rfftxXaatovv | 
loyov ^x^t ijnfp 6 F wpos ^cv ^. fa^I yap T^eoa- 
geg aQi&fiol «valoyov si<siv ot A, &, K, B, h A aqa 
itQOq TOv B tQtxXauiova Xoyov Ijjst ^keq o A ngos 
5 rov &. as di 6 A JtQog lou 0, oCtos o f npoj 
Tov i^- xal 6 A [aQtc] Jteog rbv B x^mXuaiova Xoyov 
iXEi rintQ b F aQos tov ^' oiseq edei Sei^ai. 

ly'. 

Eav a0tv oaoidijTtozovv uQiQfiol i^ijg ava- 

10 koyov, xttl noXXaTtladidGag ixaeTog iavthv 

jTot^ tfa, ot ysv6(iEvoi ii, avtmv avdXoyov 

iaovtar xal lav oC i^ ^QXVS ^ovg yevofievovg 

XoXKaitlaeiaativtsg jroKoflt Ttvas, xai awrol 

avakoyov iaovTRi [xal dsl nsgl TOtig axpov; 

16 Torto avfi^aivai]. 

"Eataisav bxotloiovv aQt&itol i|^s dvdXoyov, ol 
A, B, r, d)g b A tcqos tbv B, ofTWs o B jiqos rm' 
r, xal ol A, B, r iavzovg (ilv noXXajtlaaideavTeg 
Tows A, E, Z nouitaaav, roiig d^ A, E, Z noXXa- 
20 nXaaidaavTsg Tors H, &, K itotsCtfiiaav' Xiyto, Sti oi 
t£ A, E, Z xal ot H, &, K ilrjg dvdXoyov sietv. 

'O (i\v yaQ A tbv B itaXXaitXaataaas tov A xoici- 

ra, BXKTtQos Sl tav A, B tov A noXlaxXaeideoi 

BxdtsQov rmv M, N xoieCrta. xa\ ndXtv b (ilv B toi' 

26 r noXXaaXaatdaas tbv S TroieCra, ixdrsQos Si tav B. 

r tbv S noXXanXaaiaaag ixareQOV rav O, TI noitCta. 



1. -rgiTrlaBiova] ip- e ct 
0] mg. cp. 6. aga] om. I 
V^. IB. yivoiiivovg Y. 
aoXi.anlaaiiia€iq «-] mg. ip. 



I. as ii o A jipos tw 
11. noiti Vip. tl»M 
B. 2B. TOf A — M: 
id» P. Oj in m. 



ELEMENTORUM LIBER Vin. 309 

lam dico, esse etiam A : B '^ F^ : z^\ nam quo- 
niam quattuor numeri proportionales sunt -^, ®, K, -B, 
eTiiA:B = ^^:@^ [V def. 10]. uerum ^:® = r : z^. 
ergo j4 : B '^ r^ : z^*; quod erat demonstrandum. 

xni. 

Si quotlibet numeri deinceps proportionales sunt, 
et singuli se ipsos multiplicantes numeros aliquos 
effecerint; numeri ex iis producti proportionales erunt; 
et si numeri ab initio sumpti numeros productos mul- 
tiplicantes numeros aliquos effecerint^ hi et ipsi pro- 
portionales erunt.*) 

Sint quotlibet numeri deinceps proportionales ^, 
B, r, ita ut sit A:B = B:r,etsitAXji = ^, 
BxB = E, rxr=Z, AX^ = H, BXE = @, 
rxZ = K. dico, et numeros.-J/E, Z et H, @, K 
deinceps proportionales esse. 

A\ 1 H\ 1 

B\ 1 &\ 1 



n 1 K\ ■ 

J\ 1 iWh 



E\ 1 N\- 



Zi 1 0\ 1 

' i^ ' n ' 

I 15? 



nam sit AX B = A, Ax A = M, Bx A = N, 
et rursus sit Bxr=!S, BxS=0, TxS^^^n. 



1) Uerba aequentia %al dsi lin. 14 — avyL^aCvBi lin. 16 
subditiua uidentur; cfr. ad VII, 27. habet ea Campanus VIII, 12. 



310 



STODCBUiN ij'. 



'Ofioiios 6ii xots inava dEi^oftiv, ozt oC jd, A^ E 
xal o^ H, M, N, & egijs tieiv dvaXoyov iv ra rot J 
nQog Tov B i.6yp, xa\ hi oC E, S, Z xal o/ 0, O, 11, 
K e|^s Eletv avaloyov iv tp tov B srpog toi' F 
6 Xoya. xtti i<Sttv mq o A npog tbv B. ovroig 6 B 
jrpog lov r- xai al A, A, E Kp« rofg E, S, Z iv tm 
ttvt^ Xoya tiel xaX itt ot H, M, N, @ zoVs &, O, n, 
K. xtti ioziv tOov rb ftiv xiov A, A, E jiXij&os ta 
tav E, S, Z xX^&et, tb 6i trov H, M, iV, & tp iidi' 
10 ®, 0, 11, K' 6i' fffow cpa iatlv rog /liv 6 jj xpbs '^bv 
E, ovzag 6 E ffpog tbv Z, atg dh & H xpbg rbv 6, 
ovtms o & Kpog ro)' K' on;fp fSEi def|«(. 

td'. 
'Eav tETQiiymvos TEzgayiavov (isrpfi^ xai ij 
3 TtXEVpa lijv nlEvifiiv fictp-^aef xal iav i] nXev- 
(ftt fi}v xXevQav (xerp^, xal 6 TETgayayvos rov 
rerifdyaivov (lEtQ-^aat. 

"Eetaiaav tErQayiovoi aQt&(iol ot A, B, xXsvQal 
Si avtiov SartaOav ot F, A, b Sh A rbv B fteTQtizio' 
20 Xiyio, oti x«t o r tbv z/ (itrpEf. 

'O r yttQ rbv A KoXXttnXttettteag tbv E jtoieizm' 
01 A, E, B aga i^^s «vdXoyov Bioiv iv rp roii T 
irpoe rbv ^ Xoya. xttl ixEt oi A, E, B f|^g dvdXo- 



1. A, E] e corr. V. 2, N] e corr. V; Bupra j 

id. mg. m, 3: xal ot H, M, N, &. 3. B] Z ip. Xoyai) oon; 
ex layov q>. 5. xu^ iativ — 6: xov F] tog. ip. 7. ela' 
PB. 8. iM»] om. P. A, E] e corr. V. 10. xkI 6 

tsov P. iiiv &] ii /liv BV^. 14. Post lEipayoivos »d_ 
upifi^fiDe aupra m. 1 B 9, m. 2 V, Supra. itTeayiavav add. 
a^t^fiDf B ni. 2. IS. nXEvpa ip. 23, loyia] COrr. ex lo- 



U^ 



■ ELEMENTORUM LIBEE VUI. 3U 

iain eodem modo, quo supra'), demoiistrabimiiB, 
ni.meroa ^, A, E et H, M, N, & deinceps propor- 
tionales esse in ratione A : B, et praeterea E, S, Z 
et @, O, n, K deinceps proportionales esse in ratione 
B : r. et .4 : B = B-.r. quare etiam J, A, E et 
E, S, Z in eadem ratione sunt et praeteren U, M, N, 
& et &, O, n, K. et multitudo numerprum J, A, E 
multitudini nniQerorum E, S, Z aequalis est et mul- 
titudo numerorum H, M, N, ® multitudini numerornm 
®, O, n, K. ex aequo igitiir erit .^ : E => E : Z et 
H :& = @ : K [ VII, 14]; quod erat demoustrandum. 

XIV. 

Si numerus quadratus quadratum numerum meti- 
tur, etiam latus latua metietur; et si latus latus me- 
titur, etiam quadratus quadratum metietur. 

Sint numeri quadrati A, B, latera autem eorum 

■it^ 1 sint r, ^, et A Dumerum B me- 

B^ „ . . - . - I tiatur. dico, etiam V numerum 

Ti 1 Ji — I /i metiri. 

£,__ — .1 gjt enim rxA = E; itaque 

A, E, B deinceps proportionales sunt in ratione f ; A 
L Iprop, XI]. et quoniam A, E, B deinceps proportionales 

1) Oelttt in prop, 12, Boilioet per VU, 17 — 18. cnm 
im AxA^JeAAxS^A, erit A : B = J : A. cum 
<B=-jletBxB = E, erit A : R ^ A -. E. itaque 
^ . B ~ j3 \ A — A : E. et cum A X d ~ H, A x A = M, 

iAi J : A =• H : Mi cuin .^ x ,1 = M, B x .i = JV, erit 
! B — W ; JV = H : it;. cum Bx A^ N, B X E =~ 0, 
tA:E-^N:G-^A:li = H:M = M: N cett. 



312 



STOIXEIfiN Tj'. 




yov deiv, xal [ittQfl 6 jI tov B, ptTgEt Sfiic 

ji Tov E. xnC iativ as o ji Jtgos ihv E, oiirtss 

r Ttgog rov ^" fitrpsi aqa xuX 6 f thv ^. 

ndliv $i} 6 r zbv jd (tiTpsizm' Xeyta, oti xal b 

5 ^ TOV B [ISTQBl. 

Ttov yap avttov xttraiJxBvaa&evTojv ofioiae Ssil^o- 
Hiv, ot( of ^, E, B t§^e ttvaf.oy6v elaiv iv rp zov 
r «po^ Tov ^ i.6y^. xal inei istiv we o P sipog 
Tov -d, ovxag o A Jipos thv E, fietQtt Si o JT ti^ 
10 itf, (icT^it KQU xal ji Tov E. xai eitftv of jt^ 
B i^rfs oLvakoyoV ftBTQst aga xal o A rov B. 

'£'«1' Kpa rstpKyavog rftpayoivov ^rpjj, xcti 
;iA£vp« r^v Ttk^v^av ftfrpjjffff xkI iav ^ itXiv^a xr(v 
jiAeupKv jisTQfi, xal 6 rtrQayavog zov TET^dyiavov 

15 fltZQ-^esf 03Iep ^Ei tfit|«l. J 



I 



'Eav xvfioe aQi&fibg xv^ov aQi&fibv ftstg^: 
xal rj jtKiVQa rijv xKevQav fiizQ^SEf xal iav 
17 ^XtvQa rriv itisvgav fistpjj, xal o xvfiog zbv 

■20 XV^OV flEZQ^aBl. 

K^^og yaQ aQi9fibs b A xv^ov zbv B fiezffcA 
xal toi' fiiv A xXsvQa latca b F, zoi. de B & 
Xiyca , ori b F xbv ^ fieTQst. 

O r yaQ iavTOv JioXXajtlaeidaag zbv E itoisii 
2b di ,d iavTov itoX2.a7tkaeidaae rDi/ H aiotf^ro, 
£T( 6 r zbv /fi noXXanAaaideas xbv Z [«oiEizai], 



1. elm Y<p. 2. EJ aeq. ras. 1 litt. V. S. ftevffBt 
J] om. P, 4. Ttdliv dfj] dXla Syj iteT^riim BYtp. 

Kttl Vq!. fKTpfiria] om. BVip. 9. (iFrpsr— 10; i 

E] om. P. 10. Be«] pOBt rae. 2 litt. B. _ 13. Supra xjrp 
ytovOE et zcrgdyoivov in B sur. compp. UQi&iios et u^i^fti 



TOV 

I 



ELEMENTORUM LIBEK Vni. 313 

sunt, et A numemm B metitur, A etiam numenim 
£ metitur [prop. VII]. est autem A : E = V \ J. 
ergo etiam V numerum A metitur [VII def. 20J. 

Rursus T nomerum ^ metiatur. tlieo, etiam A 
Dumerum B metiri. 

sam iisdem comparatis similiter demonstrabimus, 
numeros A, E, B deincepa proportionales ease in 
ratione T: A. et quoniam est r : ^ = ^ : £, et r" 
nuraerum A metitur, etiam A numerum £ metitur 
[VII def. 20J. et A, E, B deinceps proportionaiea 
sunt. quare etiam A numerum B metitur.') 

Ergo si numerus quadratua quadratum numerum 
metitur, etiam latus latus metietur; et ai latus latus 
metitur, etiam quadratua quadratum metietur. 

XV. 

Si cubus numerus cubum numerum metitur, etiam 

A^ 1 latus latua metietur; et ai latua 

B( 1 latus metitur, etiam cubus cu- 

ri 1 ^' ' bum metietur. 

^[ — I Nara cubua numerus A cu- 

Ei . bum B metiatur, et numeri A 

li\ ■ 1 latna sit T, numeri B autem A. 

Z\-~ ( dico, T numerum ^ metiri. 



V 



ait enim TxT=E, ^-X/l-^-H, TxA=Z, 



1) Nam E iiamemm £ metitiir (Vll def. 30) et A 



IB. ojHp iSti afijBi] om. PB. 21. ^txQTiacim qj. 22. F] Arp, 
"1. D r] Kttl r Vqi. nfcqifici. BVip. 26. Si d lav- 

f] »nl tti o r x!>v J BVip. H] Z BVip, xal fit 6 

jov J') o 3i J lavrov BVo.. SO. Z] HBVo'. noieCio)] 

1. P. 



314 




rroisEiiiN ,)'. ^l 


TiQOS di 


Tmv r, J zov Z itoklaalaaiaeas fxarfpo^ 


TMi' 0, K nouirm. 


ipavEpov S^. oii ol E. Z, H xtti 


of J. 0, 


Ji^, B ^I^S 


avttXoyov tlatv iv rp tov f wpos 


zi)v ^ /oyp, zd:1 


insl ol A, &, K, B fl^s avdXoy6v 


5 dfiiv, xa 


Hezff{t: 


A rov B, (iszQiF liffa xal xov ft 


xai iaziv 


ros 6 ^ 


jipog zov &, ovzms V stQos zbv 


^" (l£tQB 


- apa xal 


r rov z/. m 


'AkXa 


dii (tizQiCza F tov J- liya, Sri xai^| 


A xhv B 


(let^^^osi. 


^H 


10 Tav 


yag avzcov XBZaaxsvae&dvtav ofioims ^^| 


Sei^oiiev, 


ozt ot A 


0, £*, B ii^s avdXoyov E^om^H 


■cip zov r wpog Toi' z/ Ao}'Ci}. xal imi 6 F zov^^ 


litT^et, xai iaziv 


DJS 6 r XQos xov ^, oiktos ^i^M 


ntffOq tov 


&, xal 


A apa zov & {LezQsf Sere I^H 


15 Tov B (ittgeZ 6 A 


DJiEp iSei Sst^at. ^H 


'Ekv 


. m 

TfTQaytovog a^i&nog zEtQayavov aQtl^^ 


(thv fiV 


ftrre^, 


ovdl Tj jrAswpa trjv xIevq^^ 


flETpiJOE 


[■ xav ii 


jtXsvpa zijv ^tXsvgav (li f^H 


20 rpjj, ov 
ZQ^asi. 


Si 6 TST 


ifdyavog tov Tfitpayravow (^H 




"EdTaeuv zsTeuyavoi d^i&fiol of A, B, xXev^^^ 


S'e avzmv 


letcaeav 


of r, ^, XB^ ^^7 (lEtgBita 6 A ^^H 


B- Uym 


ozi ovSh 


6 r rov z/ (iszffsl. t^M 


25 Ei yap iiBtQtZ 


& r zbv z/, (lEtp^ait xal 6 A «^l 


B. ov (lEZQdi: Si 


3 A tov B- ovSl aqa 6 V zov ^| 


[iBTQ^eH. 




6. ilai Y^. 6. e] oni. tp. {^H 


3. of] 


om. Vif. 


^fxQii «e 


«Bi i. r 


rov J] mg. m. 1 T. 9, y,etstlm3^^ 


10, «■ttov 


V- m 


om. B. 12. lov] om. P. xc(^]^H 



ELEMENTORUM LIBER VIIL 315 

r X Z — @f jd X Z = K. manifestum igitur^ nu- 
meros E, Z, H et A, @, K, B deinceps proportionales 
esse in ratione F : jd [prop. XliJ. et quoniam j4, &, 
K, B deinceps proportionales sunt^ et ji numerum B 
metitur, etiam numerum ® metitur [prop. VII]. uerum 
j4 : ® = r : ^. ergo etiam F numerum jd metitur. 
Rursus metiatur F numerum jd. dico, etiam A nu- 
merum B metiri. nam iisdem comparatis similiter 
demonstrabimuS; numeros A, 0, K, B deinceps pro- 
portionales esse in ratione F: jd. et quoniam F nu- 
merum jd metitur, et P : ^ === -^ : 0, etiam A nu- 
merum & metitur [VII def. 20]. quare etiam nume- 
rum B^) metitur A:^ quod erat demonstrandum. 

XVI. 

Si numerus quadratus quadratum numerum non 
metitur^ ne latus quidem latus metietur; et si latus 
latus non metitur; ne quadratus quidem quadratum 
metietur. 

A\ 1 Sint numeri quadrati A, B, latera 

B\ 1 autem eorum sint F, z/, et A nume- 

Pi 1 rum B ne metiatur. dico, ne F qui- 

J\ 1 dem numerum z/ metiri. 

nam si F numerum A metitur, etiam A nume- 
rum B metietur [prop. XIV]. at A numerum B non 
metitur. ergo ne F quidem numerum z/ metietur. 



1) Cfr. p. 313 not. 



2 B, om. Vqp. 19. /Ln}] snpra V. 22. dgid^iiofi m. 2 B, 

om. Vy. 23. ft^] supra V. 24. Xiyo) di P. ovd' V. 

fiAfiT^^tfeiVqo. fiSTQBi: — 26: tov J] mg..m. 1 P. 26. ovd' B. 



314 

tmv S, K MieiToi. 
(H A^ &, Kj B t|»,... 
xov ^ koya. xcu 

5 €l6lVj OCal liStQEl 

xai i6ttv off 6 yJ 
J' iiitQil aga xr * 
^Akka dri ii€T'^ 
A tbv B iistQiliOi- 
10 Tav yccQ av 
daCl^oiiBv, oti 01 
tA tov r XQog 
lietQst^ xa( i6rn 
nQog tov 9, x((' 

15 tov B liStQSt o . 



^Eav tstQ(X' 
libv iiri iistQ 
listQiqost' xR 
'jo T^jj, ovdh b 
tQti^st, 

"Earm^av r 
(U ttvrAv S6r(- 
B' ksyto^ oti 
E{ yuQ n 

B. ov llStQ! 



*:r> 



T,rjf' ji^yoj, on ovi: 

^prfis^ xai o r ror 
W &Qa o ^ rov S 



.'JOV aQi&glOV llfl lAS' 

^ .y xlsvQav (isrQij6si' 

t Mvfiov aQi^iiov rbv B 
j0 .4 nksvQa laxG^ 6 f , tov 

-j}W J ov iisrQijif€i. 
-vr J* xtt^ ^ ^ '^ov B iiS' 
. ^ ,' fov JB" ovd' &Qa 6 r 



't« 



•^'- 
-,.«' 



^ . r rbv z/* Xdyco^ ort oi?d6 
/ Tcr -^' ovd' apa 6 A rbv 



*•«?>* *"^ ^ intTtsdog TiQig 
.. w^v buoXoyov Ttksvgdv. 



3. of] om. 
.ufTi^Ci «9« xai 
tO. at'r6v qt. 



«•«■ 



>'V^ V. 



r^ 



qg. m. 1 P. 
h rxo9 



■p^ Bursus r numerum ^ ne metiatur. dico^ ne A 
By|mdem numerum B metiri. 

^^ nam si ji numerum B metitur, etiam F numerum 
~^ metietur [prop. XIV]. at F numerum ^ non me- 
titur. ergo ne A quidem numerum B metietur; quod 
^"wBt demonstrandum. 

xm 

Si cubus numerus cubum numerum nou metitur^ 
ne latus quidem latus metietur; et si latus latus non 
metitur^ ne cubus quidem cubum metietur. 

, i^ Nam cubus numerus A cubum 

1 numerum B ne metiatur, et nu- 

t 1 r meri A latus sit F, numeri B autem 

■i^ jd. dico, JTnumerum /1 non metiri. 



nam si T numerum ^ metitur, etiam A numerum 
B metietur [prop. XV]. at A numerum B non me- 
titur. ergo ne T quidem numeram ^ metitur. 

Uerum T numerum /1 ne metiatur. dico, ne A 
quidem numerum B metiri. 

nam si A numerum B metitur, etiam T numerum 
^ metietur [prop. XV]. at T numerum A non me- 
titur. ergo ne A quidem numerum B metietur; quod 
erat demonstrandum. 

xvm. 

Inter duos similes numeros planos unus medius 
est proportionalis numerus; et planus ad planum 



in ras. qp. 13. 6j (prius) corr. ex xov V. 14. ficrpcr] |»6- 
T^ijm Vqp. 16. Qv8i Vqp. 20. 6 ^] supra m. 2 V. 21. 
ftn^ ^^et de£Eat] om. BVqp. 




316 sToixEinN ■n'. 

6 A zbv B fitrp^ati. 

El yiip fiFTQcl o A xhv B, iiecff^aei. xal 6 F zov 
A. QV [i£Tp£t Si r to* ^" ovd' UQa o A xov B 

'Eav xv^og aQi&fios xvfiov api&fiov (ifj fif- 
Tp^, ovSi rj TtXEvpa tijv «ievQav fieTQ-^eei' 
x&v 7] nAcupa cijv nXEvpicv fii} fiszg^, oidi o 
I xvpog toj' xiJ/3ov fieXQ^^eei. 

Kv^og y«p apt&ftiJg 6 A xv^av «Qif^p,ov rov B 
fiij fieTQfita, xul zov (ilv A iilsvQa IffTO 6 r", tow 
di B o ^" Xiyva, ort 6 T tov A ou ftftpifffEt. 

E^ y«p fiSTQet P rov -4, xa! 6 A vov B fii- 
> rpijOEt, ou ftiiper tfi 6 ^ tui' B" ovtf' apa 6 T 

TOV d flETQEt. 

'Alla Sij fi^ fierQsiTa 6 F rbv A' Xiya, oxi ov6\ 
o A tov B fiSTQriSEi,. 

El yaQ 6 A tov B fieXQst, xal 6 f zbv ^ fU- 
3 tpijffft. oi) fkiTQet 8% T Tov A' ov6' apa o A thv 
B [lerp^esf orcrp Met det^ai. 



.Jvo ofioCcov sitmiSmv «pt#ftt5i' tlq ttscos 
avaXoyov esriv affi&fioq' xal 6 iniaESog npij 
5 tov i^ineSov Si%ka6iova Xoyov ^%et. ^ffep ^DB 
b^oXoyoq nkevQOL ngos t^v ofioAo^^ov aXevQiitiM 

1. ^ij] om. P. 3. f( yaQ nsTQfLO A rov B] mg. na. 1 P, 
y,iTQ>',att] om, P. 4. iJJ eraB. V, oi5 jierjfi 8\ h r tiv 

J] iu. 2 B. 5. ome fSti dti^ai] om. B. 9. (itte^} -j 



ELEMENTORUM LIBER Vm. 317 

Rursus r numerum ^ ue metiatur. dico^ ne A 
quidem numerum B metiri. 

nam si A numerum B metitur, etiam F numerum 
^ metietur [prop. XIV]. at F numerum ^ non me- 
titur. ergo ne A quidem numerum B metietur; quod 
erat demonstrandum. 

xm 

Si cubus numerus cubum numerum nou metitur, 
ne latus quidem latus metietur; et si latus latus non 
metitur^ ne cubus quidem cubum metietur. 

, i^ Nam cubus numerus A cubum 

1 numerum B ne metiatur, et nu- 



« 1 r meri A latus sit JT, numeri B autem 

I 1^ J. dico,rnumerum^nonmetiri. 



nam si F numerum ^ metitur, etiam A numerum 
B metietur [prop. XV]. at A numerum B non me- 
titur. ergo ne F quidem numeram ^ metitur. 

Uerum F numerum Zl ne metiatur. dico, ne A 
quidem numerum B metiri. 

nam si ji numerum B metitur, etiam F numerum 
^ metietur [prop. XV]. at F numerum ^ non me- 
titur. ergo ne A quidem numerum B metietur; quod 
erat demonstrandum. 

xvm. 

Inter duos similes numeros planos unus medius 
est proportionalis numerus; et planus ad planum 



in ras. qp. 13. 6] (prius) corr. ex rov V. 14. jicrpcr] fie- 
xQT^csi Vqp. 16. ovde Vqp. 20. 6 ^] supra m. 2 V. 21. 
onsQ ^dfi detiai] om. BVqp. 



318 



ETOIXEiaN i 



"Eazmaav 8vo ofiotoi. ixixESoi. ttQt&fiol o£ ji, B, 
xal Tcii [ilv A nhivfiai lar<oOav o£ F, ^ api&iioi, 
Totj di B ot E, Z. xal ItcsI ofioiot imTtiSoi eiatv 
o{ avakoyov i%avxB^ taq xXtvQaq, iattv apa ag i> 
b r Jipb; Tov ^, OUTOJS 6 E npos tov Z. Xiyta ovv, 
oit tmv j1, B ffg (ti'ffos dvaloyov iariv i^fiiid-fto^, xel 
o A itgbs rbv B StTiXaaiova Xoyov i%ii ^jcsq o V 
Kpog tov E 7} o ^ irpos tov Z, toirriertv ^tisq ij 
bftoXoyoq JilivQa n^bg rijv bjioXoyov [aksvQtiv^. 

10 Kal i%ei ietiv mg 6 f sipo; roi' A, ovttog o E 
ni}bq tbv Z, ivakkai, aga iatlv rag o P ^^bg tbv 
E, b ^ irpog Tov Z. xttl insl ini^sSog iartv b A, 
itXtvQixl Si avtov of F, ^, b ^ aqa tbv F noXXa- 
TtXctataattg tbv A nsTCoi-^xsv. dta tcc avtcc ff^ xal o 

15 E tbv Z jcoXXaJcXacfiaaag tov B ns^oti^xsv. b ^ 6rj 
tbv E noXkanXaataeag zbv H notsitco. xal ijtsl b 
^ tiv [isv r noXXanXaataaas tov A 'xvcoifixev , tbv 
S% E ■xoXXajiXaGideag tbv H asxoirixsv, iativ Siqk 
mg b r ngbg tbv E, ovvcos b A srpog tow H. dXX' 

20 a>s b r nQog tbv E, [outos] o A itQbs rov Z' xal 
d>s aga b A npbg tbv Z, ovtmq 6 A srpog rbv U. 
ndXtv, iitsl 6 E tbv ftev ^ noXXanXaatdaag tbv H 
ns%oii\xsv, tbv S^k Z n.oXXa%Xaeidaag tbv B neitofT)- 
xsv, Seriv aga aq b jd itpbs tbv Z, ovtcos 6 H 

35 JTpog rbv B. iStixd^rj dh xal tog 6 ^ «pos tov Z, 
ovtms o A Ttpos Toi' H' xcd ms aga o A TCpog rbv 
H, oCrmg 6 H jrpot,- tov B. of A, H, B a(fu i^ijg 



dvdXoyov tietv. 
iattv apt&ftos. 



• A, B ttQa slg (liaog dvdXoyov 



, dgi9iiai] om. V cp. 9. nlivQiiv] om. 

I. ip. 13. Jtolvnkailiaiias P. 14. nfiroi^ijMc 



i 



ELEMENTORUM LIBER Vm. 319 

duplicatam rationem habet quam latera correspon- 
dentia. 

Sint duo numeri plani similes A, Bj et latera 
numeri A sint JT, jd, numeri B autem -E, Z. et quo- 
niam similes plani numeri ii sunt^ qui latera pro- 
portionalia habent [VII def. 21], erit T: z/ = JB: Z. 
dico, inter A^ B unum medium esse proportionalem 
numerum, et esse A i B ^ F^ i E^ = A^ i Z^. 

iam quoniam est JT : z/ «= £ : Z, permutando erit 
Fi E= A \Z [Vn, 13]. et quoniam A planus est, 

A\ 1 I \r 

B\ ■ 1 I \/i 

H\ 1 I 1 E 

Zi \ 



latera autem eius JT, -^, erit ^ X r= A. eadem 
de causa erit etiam ExZ = B, iam sit AxE^H* 
et quoniam zi X F = A et Ax E = H, erit F : E 
= A : H [VII, 17]. uerum F : E = J : Z. quare 
etiam ^ : Z = A : H. rursus quoniam 

EX^ = H et EXZ = B, erit J:Z = HiB 
[Vn, 17]. demonstrauimus autem, esse etiam 

^:Z^A:H. 
quare etiam A : H = H : B. itaque A^ H, B deinceps 
proportionales sunt. ergo inter A, B unus medius 
proportionalis est numerus. 

in ras. q>. noXvnXaaidaag P. 16. noXvnXaaidaag P. 17. 
ftfii/] sapra m. 2 V. noXvnXaaidaag P. nBnoCvi%B Vqp. 18. 

noXvnXaai,daag P. 19. dX* q). 20. ovx(og] om. P. Z] 
seq. ovtmg 6 J P, del. m. 1. xal tog aqa 6 J nQog"] in 

ras. qp. 22. fisv'] om. P. noXvnXaaidaag P. 28. nsnoCrixB 
Vq>, noXvnXaaidaag P. 24. Z] in ras. tp. 28. elai Vqp. 



320 STOlXEIiiN jj". 

jidyia d^, ort xai 6 A itQog rbv B HiakasCova 
koyov l%tL ijjtap 7j oitokoyos itlBVQa %q6s t^v ofto- 
Aoyov nAfvpav, tovtiaziv ^refp 6 F in^tos rov E rj 
6 A npog Tov Z. iitel yaQ oC A, U, B i^iis iva- 
^ 2.oy6v tiaiv, 6 A apog tov B Smlaaiova loyov i%ci 
^xcQ npog roi' H. xai itsziv a>s 6 A tcqos zov H, 
ovzas t£ r irpoe zov E xal b ^ irpos zhv Z. xci 
h A ttfftt itQog zov B di.zXaaiova Xoyov ixst rjxe(f o 
r rcpoj zov E 1} 6 A jrpog rov Z- onfQ iSBi del^au 



z/uo ofioiav az£Q£av api&ftav 8vo (liaot 
avttXoyov ifiTtinTOVGiv aQt&fioi' xal 6 dT^pEOg' 
n(fbs zbv ofioiov azsQBov TQixXaaiova Xoyov 
iXEi. ^stEp ij 6^o'Aoyos itlavQa jrpoff z^v ofio- 

15 Xoyov nXavQav. 

"Eetaeav Svo oiiotot atcQeol oC A, B, xal loii 
(liv A itXtvQal iazmaav oC F, ^d, E, row Si B of 2, 
H, ®. xal ind ofioiot. azBQSoi siatv o£ avdXoyov 
ilovzES Tftg itXevQas, iaziv aqa as fiv b f XQog 

20 zbv A, ovzcas o Z Jrpog zbv H, mg Sl b ^d ;rp6s zbv 
E, ovtcog 6 H jrpog zbv &. Xiyto, Zzi. tav A, B 
6vo fiiaot dvdXoyov iiiTtiitzovaiv aQi&fioi, xal & A 
itQOS tov B zQinXaaCova Xoyov ^j;b( ^TTBp o T XQOS 
zov Z xttl 6 A jtQog tbv U xal itt & E Jtpoff zbv 8. 

25 'O r yaQ zbv ^ itoXXaitXaaidaas tbv K noiEito, 
b 6i Z tov H xoXXaaXttGidaas tbv A TtoiEitfn. 



4. Tov] zTiV P. 6. Tor] (alt.) oorr, es xi m. 2 P. 
CS« Smlaeiova ioyov h^i Ttgoe toj' B V^p. 6 T] 3 M T 
PBVy; corr. ed. Baail. Jl. (imoi] ofioioi V (corr. m. rec.}, ip. 
16. oi] apiS-fial o^ qj, Ym. 2. 17. fiiv} om. B, supra m. 



i 



ELEMENTORUM LIBER Vni 321 

lam dico, esse etiam A : B = I^ : E^ = J^ : Z\ 
nam quoniam A, H, B deinceps proportionales sunt, 
erit [V def. 9] -^ : B «= ^* : H\ 

et A:H=r:E= ^:Z. 
quare etiam A : B = F^ : E^ = ^d^ : Z*; quod erat 
demonstrandum. 

XIX. 

Inter duos similes numeros solidos duo medii 
proportionales numeri interponuntur; et solidus ad 
solidum similem triplicatam rationem habet quam 
latera correspondentia. 

Sint duo solidi similes A, B et numeri A latera 
sint JT; ^, Ej numeri jB autem Z, H, S, et quoniam 






N\- 
I 



S 



r\ 1 I z 

J\ 1 1 iH 

E\ 1 I iS 

K\ 1 

A\ 1 



M\- 



similes solidi ii sunt^ qui latera proportionalia habent 
[VII def. 21], ent r: J = Z:H, J : E = H:e. 
dico, inter A, B duos medios proportionales numeros 
interponi, et esse A:B = T^ : Z^ = ^^: H^ = E^:e^. 
sit enim rx/l = K, ZxH = A. et quoniam 



2 V. 18. d^i^^ol ot Vqp. 19. (ihv 6] 6 (kh Vqp, o B. 

24. uaQ (prius) om. B, mg. rj. iti] iari tp. 

Enolides, edd. Heiberg et Meoge. II. 21 



322 



STOIXEIliN ij'. 



ixel ot r, d xotq Z, H iv rp uvTa Xoya siaiv, xel 
ix iiiv xmv r, ^ iaxiv o K. In &% Ttav Z, U o 4. 
ol K, A [«pa] oftotot iniTtiHoi ticiv aQi&noi' rmv A'. 
A KQct tig (lioog dvtiloyov iexiv UQi&iiog. ieTto « 
5 Af, 6 M aga iatlv o ix xmv ^, Z, {og iv tco Sfii 
rovToti &£D)Q^iiaTi iStix^^' ""^ ^^^^ ^ ^ "^ov ^iv 
r nollaTtkaeiaOus rov K atnoitiXEv , zov Sh Z noi- 
A«jrAKff(ne«S xov M ntnoiijxtv, lati.v apa tog 6 F 
jrpog xbv Z, orTog 6 K repog rov M. aX,l' ^g 6 K 

10 «pog rov M, 6 M n^{tg xov A. of K, M, A «pff 
l|^g tiaiv dvdXoyov iv rm rov F TtQog zov Z X6- 
ya. xal ixti iaxiv log o F JEpog zov ^, ovrmg o 
Z ngog rov H, ivalla^ apa iexlv mg o T JtQog roi' 
Z, otiTws 6 ^ wpoj xhv H. Sia xd avxa Si, xal 

15 rag 6 ^ TtQog xbv H, ovtio? 6 E icpog tbv &. oi 
K, M, A Spo: i^r^g tloiv avaXoyov ev xe rp xov F 
Jtpoj xov Z i,6yo3 xal xa xov A Jipog xov H xal 
ttt xa Tou E ngos tov &. ixattQog ^17 xmv E, & 
xbv M xoi.kaaXaaicceas sxatspov rmv N, S jrotftio. 

20 xal insl axtQsog iariv 6 A, ^Xtvpal Sh avxov tCOtv 
of r, A, E, o E Sga xbv ix xmv F, J nolkanka- 
aiaaag xov A ntnoiy}xtv. b Si ix xmv F, z/ ieriv 
K' 6 E aqa xov K aokXaitXaeiaaag xov A ntTtoiijxev. 
Sia xa avxa Sri xal 6 ® tbv A noXlanKaOidoag xbv 

25 B ntjtoiijxtv. xal inel o E xbv K nolianXaataaits 
xbv A ntTtoixixsv , dXXa fiijv xal xbv M xoXXanXa- 
aideag xbv JV ntnoiyjxtv, tativ dqa mg b K affbs 
xov M, ovtnag b A ngbg xbv N. ^g Sl 6 K ngbg 
xbv M, ovxmg xt F rcpos xbv Z xal b ^ Jipog 

80 101' H xal ixi 6 E stQog xbv &' xal cbg aQO 
1. of] corr. es 6 m. 2 P. ilei Vqs, 3. Sffo.] oiq. P. 



ELEMENTORUM UEER Vlir. 323 

r, ^ et Z, H in eadem ratione sunt, et T x zf = JST, 
Z-X. H = A, numeri K, A aimiles plani Bunt [VII 
def, 21]. itaque inter K, A vmua mediua est pro- 
portionalis numerua [prop. XVIII], ait M. itaque 
JVf = ^ X Z, ut in propositione praeeedenti demon- 
stratum est [p. 318, 15; 2G]. et quoniam 

<^xr= «■ et .^XZ = JW, erit r:Z=- A".M 
[Vn, 17]. uernm A: : M= M:^. itaque ff, itf, ,^ 
deinceps proportionales aunt in ratione F : Z. et 
quoniam eat F: ^ = Z : H, permutando erit 

r:Z = J:H [VII, 13]. 
eadem de causa erit etiam ^ : H = E : @. itaque 
K, M, A deineeps proportionales sunt in rationibus 
r:Z, ^:H, E:@. ia.tu sit ExM ^ N et &x M=S. 
et quoniam A solidus est, et latera eius sunt r, ^, E, 
erit EX Vx J = A. uerum Fx ^ = K. itaque 
E X K = A. eadem de causa etiam @ x A = B. 
et quoniam E x K = A, et E x M = N, erit 
K:M=A:N [VII, 17]. uerum 

K:M=r:Z = ^:JJ = £:®. 



S. Post iitlt^i) add. Y^: ^ntv Sga (hi ip) as o K Vfie t&v 
M, o M ttfot tbv A\ idem B mg. ^. 2. 7, nsnr>l-r\%e Vip. 

B. oU' mt 6 K flfpos 10» M] mg. ip. 10. 6 ] ovrios & Vtp, 

11. eleiv] om. P, supra m. 1 V. 14, Sia tcc avta 9n xai] P; 
xu3.iv ixei IdTiv tos 6 iJ agos lov B, oviois □ H ngos tov S, 
ivaHii UQa iaziv Theon (BVqi). 16. K, A, MY^. ai)a\ 
Ixl f. avttloyov iluiv Vqp. 17. Xoym] om. V^. Tp] 

om.Vqo. 21. r] (priuB) eraa, V. 22. J] seq.in P: «oUonJlw- 
giaaag, Eed delet. 23. ncnolrivit Y f. 24. Poet itollaxXu- 



i 



na«as add, Theon: xov i* t6v Z, H (BVip), 25. TtenoiTiKt 
V^. 30. M i:»"- ei on m. 1 P; for 
&S] a>S BY^J. 



324 



STOIXEIilN 7,'. 



r wpos roi' Z ml o ^ XQog rov H xal 6 E itpog tri» 
©, ovTog 6 j4 xqos tbv N. ndXiv, ixei ixazfQog 
Twi' E, & tov M noi.lanXaeta6as ixdrcQ